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Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

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7. RESOLUTION OF VECTORS 

If Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET be any two non-zero vectors in a plane with different directions and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET be another vector in the same plane. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET can be expressed as a sum of two vectors-one obtained by multiplying by a real number and the other obtained by multiplying by another real number.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (where l and m are real numbers)

We say that Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has been resolved into two component vectors namely

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (where l and m are real number)

We say that Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has been resolved into two component vectors namely

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET respectively. Hence one can resolve a given vector into two component vectors along a set of two vectors - all the three lie in the same plane.

7.1 Resolution along rectangular component : 

It is convenient to resolve a general vector along axes of a rectangular coordinate system using vectors of unit magnitude, which we call as unit vectors. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET are unit along x, y and z-axis as shown in figure below :

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

7.2 Resolution in two Dimensions 

Consider a vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET that lies in xy plane as shown in figure,

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ⇒ Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The quantities Ax and Ay are called x-and y-components of the vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.

Ax is itself not a vector but Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is a vector and so it Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.

Ax = A cosθ and Ay = A sinθ

It's clear from above equation that a component of a vector can be positive, negative or zero depending on the value of q. A vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET can be specified in a plane by two ways :

(a) its magnitude A and the direction q it makes with the x-axis; or

(b) its components Ax and Ay            A = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, θ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Note : If A = Ax ⇒ Ay = 0 and if A = Ay ⇒ Ax = 0 i.e., components of a vector perpendicular to itself is always zero. The rectangular components of each vector and those of the sum Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET are shown in figure.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

We saw that

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is equivalent to both

Cx = Ax + Bx

and Cy = Ay + By

Refer figure (b)

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has been resolved in two axes x and y not perpendicular to each other. Applying sine law in the triangle shown, we have

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

or Rx = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Ry = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

If α+β = 90°, Rx = R sinβ and Ry = R sin

Ex.7 Resolve the vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along an perpendicular to the line which make angle 60° with x-axis. 

 

Sol.Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

so the component along line = |Ay cos 30° + Ax cos 60°| and perpendicular to line = |Ax sin 60° - Ay sin 30°|

Ex.8 Resolve a weight of 10 N in two directions which are parallel and perpendicular to a slope inclined at 30° to the horizontal 

Sol. Component perpendicular to the plane

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

= Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET N

and component parallel to the plane

W|| =W sin 30° = (10) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = 5 N

Ex.9 Resolve horizontally and vertically a force F = 8 N which makes an angle of 45° with the horizontal. 

Sol. Horizontal component of is          

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

FH = F cos 45° = (8) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                   

and vertical component of is

Fv = F sin 45° = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETAns.

 8. PROCEDURE TO SOLVE THE VECTOR EQUATION

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ...(1)

(a) There are 6 variables in this equation which are the following :

(1) Magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and its direction

(2) Magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and its direction

(3) Magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and its direction.

(b) We can solve this equation if we know the value of 4 variables [Note : two of them must be directions]

(c) If we know the two direction of any two vectors then we will put them on the same side and other on the different side.

For example 

If we know the directions of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET direction is unknown then we make equation as follows:-

 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(d) Then we make vector diagram according to the equation and resolve the vectors to know the unknown values.

Ex.10 Find the net displacement of a particle from its starting point if it undergoes two successive displacements given by Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, 37° North of West, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, 53° North of East 

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Angle from west - east axis (x - axis)
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.11 Find magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and direction of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET . If Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET makes angle 37° and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET makes 53° with x axis and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has magnitude equal to 10 and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has 5. (given Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET) 

Sol.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

B = 5 (magnitude can not be negative) & Angle made by A
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.12 Find the magnitude of F1 and F2. If F1, F2 make angle 30° and 45° with F3 and magnitude of F3 is 10 N. (given Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET) 

Sol. 

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

9. SHORT-METHOD

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

If their are two vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and their resultant make an angle α with Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. then A sin α = β sin β

Means component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET perpendicular to resultant is equal in magnitude to the component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET perpendicular to resultant.

Ex.13 If two vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET make angle 30° and 45° with their resultant and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has magnitude equal to 10, then find magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Sol. B sin 60° = A sin 30°   

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                                

⇒ 10 sin 60° = A sin 30°

⇒ A = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.14 If Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET have angle between them equals to 60° and their resultant make, angle 45° with Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET have magnitude equal to 10. Then Find magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. 

Sol. here a = 45° and b = 60° -45° = 15°

so A sinα = B sinβ

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

10 sin 45° = B sin 45°

So B = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

= Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

10. ADDITION AND SUBTRACTION IN COMPONENT FORM :

Suppose there are two vectors in component form. Then the addition and subtraction between these two are

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Also if we are having a third vector present in component form and this vector is added or subtracted from the addition or subtraction of above two vectors then

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Note : Modulus of vector A is given by

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.15 Obtain the magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET if 

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET      and         Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET Magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

= Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETAns. 

Ex.16 Find Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET if Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET make angle 37° with positive x-axis and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET make angle 53° with negative x-axis as shown and magnitude of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is 5 and of B is 10.

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

for Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    +   Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    =     Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

so the magnitude of resultant will be = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

and have angle θ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET from negative x - axis towards up

for Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

So the magnitude of resultant will be

= Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

and have angle Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET from positive x-axis towards down.

11. MULTIPLICATION OF VECTORS (The Scalar and vector products) : 

11.1 Scalar Product 

The scalar product or dot product of any two vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, denoted as Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (read Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET dot Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET) is defined as the product of their magnitude with cosine of angle between them.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Thus,

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (here θ is the angle between the vectors)

Properties : 

  • It is always a scalar which is positive if angle between the vectors is acute (i.e.< 90°) and negative if angle between them is obtuse (i.e., 90° < q £ 180°)
  • It is commutative i.e. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • It is distributive, i.e. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • As by definition Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = AB cosθ . The angle between the vectors θ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Geometrically, B cosθ is the projection of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET onto Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and vice versa

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = B cosθ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (Projection of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET on Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET)

Component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = A cosθ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (Projection of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET on Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET)

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

  • Scalar product of two vectors will be maximum when cosθ = max = 1, i.e., θ = 0°,

             i.e., vectors are parallel ⇒ Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

  • If the scalar product of two non-zero vectors vanishes then the vectors are perpendicular.
  • The scalar product of a vector by itself is termed as self dot product and is given by

           Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = AA cosθ = A2Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

  • In case of unit vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET,  
    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
    In case of orthogonal unit vectors,Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                                                                                                     

 Ex.17 If the vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET are perpendicular to each other. Find the value of a? 

Sol. If vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET are perpendicular

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET     ⇒   Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

⇒ a2 -2a -3 = 0      ⇒   a2 -3a a -3 = 0

⇒ a(a -3) +1 (a -3 )    ⇒      a = -1, 3

Ex.18 Find the component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ?

Sol. Component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is given by Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET hence required component

= Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.19 Find angle between Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ? 

Sol. We have cosθ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

cosθ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                    θ = cos-1Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.20 (i) For what value of m the vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is perpendicular to Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(ii) Find the component of vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along the direction of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ? 

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(i) m = -10 (ii) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Important Note : 

Components of b along and perpendicular to a.

Let Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET . Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET represent two (non-zero) given vectors a, b respectively. Draw BM perpendicular to Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

From ΔOMB, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Thus Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET are components of b along a and perpendicular to a.

Now
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Hence, components of b along a perpendicular to a are.
(a . b/ |a|2) a and b - (a . b / |a|2) a respectively.

Ex.21 The velocity of a particle is given by Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. Find the vector component of its velocity parallel to the line Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.

Sol. Component of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET along Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

11.2 Vector product 

The vector product or cross product of any two vectors and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, denoted as

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (read Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET cross Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET) is defined as :

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Here θ is the angle between the vectors and the direction Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is given by the right - hand - thumb rule.

Right - Hand - Thumb Rule : 

To find the direction of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, draw the two vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET with both the tails coinciding. Now place your stretched right palm perpendicular to the plane of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET in such a way that the fingers are along the vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and when the fingers are closed they go towards Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. The direction of the thumb gives the direction of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.  

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Properties : 

  • Vector product of two vectors is always a vector perpendicular to the plane containing the two vectors i.e. orthogonal to both the vectors and , though the vectors and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET may or may not be orthogonal.
  • Vector product of two vectors is not commutative i.e. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET But Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • The vector product is distributive when the order of the vectors is strictly maintained i.e.Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • The magnitude of vector product of two vectors will be maximum when sinθ = max = 1. i.e. θ = 90°                                                                                                                 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • The magnitude of vector product of two non-zero vectors will be minimum when |sinθ| = minimum = 0, i.e., θ = 0° or 180° and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET i.e., if the vector product of two non-zero vectors vanishes, the vectors are collinear.
  • The self cross product i.e. product of a vector by itself vanishes i.e. is a null vector.Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • In case of unit vector Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ⇒ Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • In case of orthogonal unit vectors Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET in accordance with right-hand-thumb-rule,
    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  • In terms of components. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
     

Ex.22 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is East wards and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is downwards. Find the direction of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET × Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ? 

Sol. Applying right hand thumb rule we find that Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is along North.

Ex.23 If Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, find angle between Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET AB cosθ = AB sinθ            tanθ = 1                ⇒ θ = 45°

Ex.24 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ⇒ Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET here Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is perpendicular to both Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Ex.25 Find Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET if Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.26 (i) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is North-East and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is down wards, find the direction of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(ii) Find Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET × Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET if Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Ans. (i) North - West.                 (ii) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

12. POSITION VECTOR : 

Position vector for a point is vector for which tail is origin & head is the given point itself.

Position vector of a point defines the position of the point w.r.t. the origin.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

13. DISPLACEMENT VECTOR :

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Change in position vector of particle is known as displacement vector.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Thus we can represent a vector in space starting from (x , yj & ending at

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

CALCULUS 

14. Constants : They are fixed real number which value does not change

Ex. 3, e, a, -1, etc.

15. Variable : 

Something that is likely to vary, something that is subject to variation.

or

A quantity that can assume any of a set of value.

Types of variables. 

(i) Independent variables : Independent variables is typically the variable being manipulated or changed

(ii) dependent variables : The dependent variables is the object result of the independent variable being manipulated.

Ex. y = x2

here y is dependent variable and x is independent variable

16. FUNCTION : 

Function is a rule of relationship between two variables in which one is assumed to be dependent and the other independent variable.

The temperatures at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). Here elevation above sea level is the independent & temperature is the dependent variable.

The interest paid on a cash investment depends on the length of time the investment is held. Here time is the independent and interest is the dependent variable.

In each case, the value of one variable quantity (dependent variable), which we might call y, depends on the value of another variable quantity (independent variable), which we might call x. Since the value of y is completely determined by the value of x, we say that y is a function of x and represent it mathematically as y = f(x).

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

all possible values of independent variables (x) are called domain of function.

all possible values of dependent variable (y) are called Range of function.

Think of function f as a kind machine that produces an output value f(x) in its range whenever we feed it an input value x from its domain (figure).

When we study circles, we usually call the area A and the radius r. Since area depends on radius, we say that A is a function of r, A = f(r). The equation A = πr2 is a rule that tells how to calculate a unique (single) output value of A for each possible input value of the radius r.

A = f(x) = πr2. (Here the rule of relationship which describes the function may be described as square & multiply by π)

if    r = 1   A = π

if    r = 2   A = 4π

if    r = 3    A = 9π

The set of all possible input values for the radius is called the domain of the function. The set of all output values of the area is the range of the function.

We usually denote functions in one of the two ways :

1. By giving a formula such as y = x2 that uses a dependent variable y to denote the value of the function.

2. By giving a formula such as f(x) =x2 that defines a functions symbols f to name the function.

Strictly speaking, we should call the function f and not f(x).

y = sin x. Here the function is y since, x is the independent variable.

Ex.27 The volume V of ball (solid sphere) of radius r is given by the function V(r) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The volume of a ball of radius 3m is? 

Sol. V(3) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = 36 pm3.

Ex.28 Suppose that the function F is defined for all real numbers r by the formula. 

F(r) = 2 (r -1) +3. 

Evaluate F at the input values 0, 2 x 2, and F(2). 

Sol. In each case we substitute the given input value for r into the formula for F:

F(0) = 2(0 -1) + 3 = -2 + 3 = 1

F(2) = 2(2 -1) + 3 = 2 + 3 =5

F(x + 2) = 2 (x + 2 -1) + 3 = 2x + 5

F(F(2)) = F(5) = 2(5 -1) 3 = 11

 

Ex. 29 function f(x) is defined as f(x) = x2 + 3, Find f(0), f(l), f(x>), f(x + 1) and f(f(l))

Sol.  f(0) = 02 + 3  = 3
f(1)  =  l2 + 3 = 4
f(x2) =  (x2)2 +3  = x4 + 4
f(x +1)  =  (x + 1)2  + 3   = x2 + 2x + 4
= f(4)  = 42+3  = 19

17. Differentiation

Finite difference : 

The finite difference between two values of a physical is represented by Δ notation.

For example :

Difference in two values of y is written as Δy as given in the table below.

y2

100

100

100

y1

50

99

99.5

Δy = y2 - y1

50

1

0.5


Infinitely small difference : 

The infinitely small difference means very-very small difference. And this difference is represented by 'd' notation instead of 'D'.

For example infinitely small difference in the values of y is written as 'dy'

if y2 = 100 and y1 = 99.9999999999999.....

then dy = 0.00000000000000..........00001

Definition of differentiation 

Another name of differentiation is derivative. Suppose y is a function of x or y = f(x)

Differentiation of y with respect to x is denoted by symbols f' (x)

where f'(x) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET; dx is very small change in x and dy is corresponding very small change in y.

Notation : There are many ways to denote the derivative of function y = f(x), the most common notations are these :
 

y

"y prime"

Nice and brief and does not name the independent variable

dy/dx

" dy by dx"

Names the variables and uses d for derisive

df/dx

" df by dx"

Emphasizes the function's name

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

” d by dx of f"

Emphasizes the idea that differentiation is an operation performed on f.

Dxf

" dx of f"

A common operator notation

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

” y dot"

One of Newton's notations, now common for time derivative i.e. dy/dt

 

Average rates of change : 

Given an arbitrary function y = f(x) we calculate the average rate of change of y with respect to x over the interval (x, x +Δx) by dividing the change in value of y, i.e., Dy = f(x+Δx) -f(x), by length of interval Δx over which the change occurred.

The average rate of change of y with respect to x over the interval [x, x+Δx]

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Geometrically

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = tanθ = Slope of the line PQ

In triangle QPR tanθ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

therefore we can say that average rate of change of y with respect to x is equal to slope of the line joining P & Q.

The derivative of a function 

We know that Average rate of change of y w.r.t x is -

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

If the limit of this ratio exists as Δx → 0, then it is called the derivative of given function f(x) and is denoted as

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

18. GEOMETRICAL MEANING OF DIFFERENTIATION : 

The geometrical meaning of differentiation is very much useful in the analysis of graphs in physics. To understand the geometrical meaning of derivatives we should have knowledge of secant and tangent to a curve.

Secant and Tangent to a Curve 

Secant : - A secant to a curve is a straight line, which intersects the curve at any two points.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Tangent : 

A tangent is a straight line, which touches the curve a particular point. Tangent is limiting case of secant which intersects the curve at two overlapping points.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

In the figure - 1 shown, if value of Δx has gradually reduced then the point Q will move nearer to the point P. If the process is continuously repeated (Figure-2) value of Δx will be infinitely small and secant PQ to the given curve will become a tangent at point P.

Therefore

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

we can say that differentiation of y with respect to x, i.e. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is equal to slope of the tangent at point P (x,y)

or tanθ = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(From fig-1 the average rate change of y from x to x+Δx is identical with the slope of secant PQ)

Rule No. 1 Derivative Of A Constant

The first rule of differentiation is that the derivative of every constant function is zero.

If c is constant, then Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.30Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Rule No.2 Power Rule

If n is a real number, then Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

To apply the power Rule, we subtract 1 from the original exponent (n) and multiply the result by n.

Ex.31Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Function defined for x > 0 derivative defined only for x > 0
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Function defined for x > 0 derivative not defined at x = 0

Rule No.3 The Constant Multiple Rule

If u is a differentiable function of x, and c is a constant, then Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

In particular, if n is a positive integer, then Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.34 The derivative formula 

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

says that if we rescale the graph of y = x2 by multiplying each y-coordinate by 3, then we multiply the slope at each point by 3. 

Ex.35 A useful special case

The derivative of the negative of a differentiable function is the negative of the function's derivative. Rule 3 with c = -1 gives.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Rule No.4 The Sum Rule

The derivative of the sum of two differentiable functions is the sum of their derivatives.

If u and v are differentiable functions of x, then their sum u+v is differentiable at every point where u and v are both differentiable functions in their derivatives.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The sum Rule also extends to sums of more than two functions, as long as there are only finite functions in the sum. If u1, u2, ........ un is differentiable at x, then so if u1+u2 ....... +un, then

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

 Notice that we can differentiate any polynomial term by term, the way we differentiated the polynomials in above example.

Rule No. 5 The Product Rule 

If u and v are differentiable at x, then if their product uv is considered, then Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.

The derivative of the product uv is u times the derivative of v plus v times the derivative of u. In prime notation

(uv)' = uv' + vu'.

While the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product of two functions is not the product of their derivatives. For instance,

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    while Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, which is wrong

Ex.37 Find the derivatives of y = (x2+1) (x3+3)  

Sol. Using the product Rule with u = x2+1 and v = x3+3, we find

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = (x2+1) (3x2) + (x3+3) (2x)

= 3x4 + 3x2 + 2x4 + 6x = 5x4 + 3x2 + 6x

Example can be done as well (perhaps better) by multiplying out the original expression for y and differentiating the resulting polynomial. We now check :

y = (x2 + 1) (x3 + 3) = x5 + x3 + 3x2 + 3

 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = 5x4 + 3x2 + 6x

This is in agreement with our first calculation.

There are times, however, when the product Rule must be used. In the following examples. We have only numerical values to work with.

Ex.38 Let y = uv be the product of the functions u and v. Find y'(2) if u(2) = 3, u'(2) = -4, v(2) = 1, and v'(2) = 2. 

Sol.

From the Product Rule, in the form y' = (uv)' = uv' + vu',
we have y'(2) = u(2) v'(2) + v(2) u'(2)
= (3) (2) + (1) (-4) = 6-4 = 2

Rule No.6 The Quotient Rule

If u and v are differentiable at x, and v(x) ¹ 0, then the quotient u/v is differentiable at x,

and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Just as the derivative of the product of two differentiable functions is not the product of their derivatives, the derivative of the quotient of two functions is not the quotient of their derivatives.

Ex.39 Find the derivative of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Sol. We apply the Quotient Rule with u = t2 -1 and v = t2 1

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET           Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Rule No. 7 Derivative Of Sine Function

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.40 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Rules No.8 Derivative Of Cosine Function

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.41 (a) y = 5x + cos x           Sum Rule

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    Product Rule
 = sin x(— sin x) + cos x (cos x)
 = cos2 x - sin2 x - cos 2x

Rule No. 9 Derivatives Of Other Trigonometric Functions 

Because sin x and cos x are differentiable functions of x, the related functions

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ;    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ;    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

are differentiable at every value of x at which they are defined. There derivatives, Calculated from the Quotient Rule, are given by the following formulas.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ;   Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ;    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.42 Find dy / dx if y = tan x. 

Sol.Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex. 43 
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Rule No. 10 Derivative Of Logarithm And Exponential Functions

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ,                        Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.44 y = ex . loge (x)

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET   ⇒   Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Rule No. 11 Chain Rule Or `Outside Inside' Rule

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

It sometimes helps to think about the Chain Rule the following way. If y = f(g(x)),

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = f'[g(x)] . g'(x)

In words : To find dy/dx, differentiate the "outside" function f and leave the "inside" g(x) alone; then multiply by the derivative of the inside.

We now know how to differentiate sin x and x2 -4, but how do we differentiate a composite like sin(x2 -4)?

The answer is, with the Chain Rule, which says that the derivative of the composite of two differentiable functions is the product of their derivatives evaluated at appropriate points. The Chain Rule is probably the most widely used differentiation rule in mathematics. This section describes the rule and how to use it. We begin with examples.

Ex.45 The function y = 6x -10 = 2(3x -5) is the composite of the functions y = 2u and u = 3x -5. How are the derivatives of these three functions related ?

Sol. We have Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Since 6 = 2 × 3 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Is it an accident that Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET ?

If we think of the derivative as a rate of change, our intuitions allows us to see that this relationship is reasonable. For y = f(u) and u = g(x), if y changes twice as fast as u and u changes three times as fast as x, then we expect y to change six times as fast as x.

Ex.46 Let us try this again on another function. 

y = 9x4 +6x2 +1 = (3x2 +1)2

is the composite y = u2 and u = 3x2 + 1. Calculating derivatives. We see that

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = 2 (3x2 + 1). 6x = 36x3 + 12 x

and Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = 36 x3 + 12 x

Once again, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The derivative of the composite function f(g(x)) at x is the derivative of f at g(x) times the derivative of g at x.

Ex.47 Find the derivation of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Sol. Here y = f(g(x)), where f(u) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and u = g(x) = x2 + 1. Since the derivatives of f and g are

f' (u) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET and g'(x) = 2x,

the Chain Rule gives

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = f' (g(x)).g'(x) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.g'(x) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. (2x) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.48

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex. 49   Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET u = 1 - x2 and n = 1/4
   (Function defined) on [-1, 1]

  Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Rule No. 12 Power Chain Rule 

If Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.50 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = -1 (3x -2)-2Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

= -1 (3x -2)-2 (3) = -Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

In part (d) we could also have found the derivation with the Quotient Rule.

Ex.51 (a)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Sol. Here u = Ax B, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(b)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
(c)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETlog(Ax B) = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.A

(d)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETtan (Ax + B) = sec2 (Ax + B).A
(e)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Note : These results are important

19. DOUBLE DIFFERENTIATION 

If f is differentiable function, then its derivative f' is also a function, so f' may have a derivative of its own, denoted by Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. This new function f'' is called the second derivative of because it is the derivative of the derivative of f. Using Leibniz notation, we write the second derivative of y = f(x) as

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Another notation is f''(x) = D2 f(x).

Ex.52 If(x) = x cos x, find f'' (x)

Sol. Using the Product Rule, we have f'(x)   Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

To find f" (x) we differentiate f'(x)

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

= - x cos x - sinx - sinx = - x cos x - 2 sin x

20. Application of derivative Differentiation as a rate of change

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is rate of change of 'y' with respect to 'x' :

For examples :

(i) v = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET this means velocity 'v' is rate of change of displacement 'x' with respect to time 't'

(ii) a = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET this means acceleration 'a' is rate of change of velocity 'v' with respect to time 't'.

(iii) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET this means force 'F' is rate of change of momentum 'p' with respect to time 't'.

(iv) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET this means torque 't' is rate of change of angular momentum 'L' with respect to time 't'

(v) Power = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET this means power 'P' is rate of change of work 'W' with respect to time 't'

Ex.53 The area A of a circle is related to its diameter by the equation Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET.

How fast is the area changing with respect to the diameter when the diameter is 10 m ?

Sol. The (instantaneous) rate of change of the area with respect to the diameter is

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

When D =10m, the area is changing at rate (π/2) = 5π m2/m. This means that a small change ΔD m in the diameter would result in a change of about 5p ΔD m2 in the area of the circle.

Physical Example : 

Ex.54 Boyle's Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant : PV = C. Find the rate of change of volume with respect to pressure. 

Sol. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.55 (a) Find the average rate of change of the area of a circle with respect to its radius r as r changed from 

(i) 2 to 3                      (ii) 2 to 2.5                     (iii) 2 to 2.1 

(b) Find the instantaneous rate of change when r = 2. 

(c) Show that there rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically when this is true by drawing a circle whose radius is increased by an amount Δr. How can you approximate the resulting change in area ΔA if Δr is small ? 

Sol. (a) (i) 5π (ii) 4.5 π (iii) 4.1 π

(b) 4π

(c) ΔA ≈ 2 πrΔr

21. MAXIMA & MINIMA

Suppose a quantity y depends on another quantity x in a manner shown in figure. It becomes maximum at x1 and minimum at x2. At these points the tangent to the curve is parallel to the x-axis and hence its slope is tanθ = 0. Thus, at a maximum or a minima slope

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Maxima 

Just before the maximum the slope is positive, at the maximum it is zero and just after the maximum it is negative. Thus, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET decrease at a maximum and hence the rate of change of Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is negative at a maximum i.e., Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET at maximum. The quantity Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is the rate of change of the slope. It is written as Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. Conditions for maxima are : (a) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (b) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Minima 

Similarly, at a minimum the slope changes from negative to positive, Hence with the increases of x. The slope is increasing that means the rate of change of slope with respect to x is positive.

Hence Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Conditions for minima are : 

(a) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                  (b) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Quite often it is known from the physical situation whether the quantity is a maximum or a minimum. The test on Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET may then be omitted.

Ex.56 Find maximum or minimum values of the functions : 

(A) y = 25x2 + 5 -10x          (B) y = 9 -(x -3)2 

Sol. (A) For maximum and minimum value, we can put Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

or Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                     Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET x = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Further, Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

or Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET has positive value at x = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. Therefore, y has minimum value at x = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. Therefore, y has minimum value at x = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET. Substituting x =Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET in given equation, we get

ymin = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(B) y = 9 -(x -3)2 = 9 -x2 +-9 6x

or y = 6x -x2

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

For minimum or maximum value of y we will substitute Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

or 6 -2x = 0

x = 3

To check whether value of y is maximum or minimum at x = 3 we will have to check whether Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is positive or negative.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

or Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is negative at x = 3. Hence, value of y is maximum. This maximum value of y is,

ymax = 9 -(3 -3)2 = 9

22. INTEGRATION 

Definitions : 

A function F(x) is a antiderivative of a function f(x) if

F'(x) = f(x)

for all x in the domain of f. The set of all antiderivatives of f is the indefinite integral of f with respect to x, denoted by

 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The symbol Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET is an integral sign. The function f is the integrand of the integral and x is the variable of integration.

For example f(x) = x3 then f'(x) = 3x2

So the integral of 3x2 is x3

Similarly if f(x) = x3 + 4

there for the general integral of 3x2 is x3 + c where c is a constant

One antiderivative F of a function f, the other antiderivatives of f differ from F by a constant. We indicate this in integral notation in the following way :

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                 .....(i)

The constant C is the constant of integration or arbitrary constant, Equation (1) is read, "The indefinite integral of f with respect to x is F(x) + C." When we find F(x) + C, we say that we have integrated f and evaluated the integral.

Ex.57 Evaluate Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Sol.Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The formula x2 + C generates all the antiderivatives of the function 2x. The function x2+ 1, x2 -π, and

x2+Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET are all antiderivatives of the function 2x, as you can check by differentiation.

Many of the indefinite integrals needed in scientific work are found by reversing derivative formulas.

Integral Formulas

Indefinite Integral    Reversed derivated formula

1. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET,n ¹ -1, n rational    

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (special case)

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = xn

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

2. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET  Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
3. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
4. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET  Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
5. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET(-cot x) = cosec2 x
6. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEETResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = sec x tan x
7. Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = -cosec x +CResolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET(-cosec x) = cosec x cot x

 Ex.58 Examples based on above formulas :

(a) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(b) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                                                                    Formula 1 with n = 5

(c) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                           Formula 1 with n = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(d) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                                                      Formula 2 with k = 2

(e) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET     Formula 3 with k = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.59 Right :Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = x sin x + cos x C

Reason : The derivative of the right-hand side is the integrand :

Check : Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = x cos x + sin x -sin x + 0 = x cos x.

Wrong :Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = x sin x +C

Reason : The derivative of the right-hand side is not the integrand :

Check :Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = x cos x + sin x + 0 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET x cos x

 

Rule No. 1 Constant Multiple Rule

  • A function is an anti derivative of a constant multiple k of a function f if and only if it is k times an antiderivative of f.

                 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Ex.60 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

 

Rule No.2   Sum And Difference Rule

  • A function is an anti derivative of a sum or difference f ± g if and only if it is the sum or difference of an anti derivative of f an anti derivative of g.

                 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET               
Ex.61 Term-by-term integration 

Evaluate : Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Sol. If we recognize that (x3/3) -x2 +5x is an anti derivative of x2 -2x +5, we can evaluate the integral as

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

If we do not recognize the anti derivative right away, we can generate it term by term with the sum and difference Rule :

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

This formula is more complicated than it needs to be. If we combine C1, C2 and C3 into a single constant

C = C1 + C2 + C3, the formula simplifies to

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

and still gives all the anti derivatives there are. For this reason we recommend that you go right to the final form even if you elect to integrate term by term. Write
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Find the simplest anti derivative you can for each part add the constant at the end.

Ex.62 We can sometimes use trigonometric identities to transform integrals we do not know how to evaluate into integrals. The integral formulas for sin2 x and cos2 x arise frequently in applications.

(a) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET   = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
  = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(b) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET  Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET As in part (a), but with a sign change

23. Some Indefinite integrals (An arbitrary constant should be added to each of these integrals. 

(a) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET (provided n ¹ --1) C    
(b) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(c) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 
(d) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET
(e) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET   
(f)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 

Ex.63 (a) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 
(b)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(c) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 
(d)Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(e) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 
(f) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(g) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET     
(h) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

24. DEFINITE INTEGRATION OR INTEGRATION WITH LIMITS 

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.64 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET  = 3 [4 -(-1)] = (3) (5) = 15

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET    = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET + cos (0) = -0 + 1 = 1

Ex.65 (1) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(2) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

(3) Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

25. APPLICATION OF DEFINITE INTEGRAL 

Calculation Of Area Of A Curve. 

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

From graph shown in figure if we divide whole area in infinitely small strips of dx width.

We take a strip at x position of dx width.

Small area of this strip dA = f(x) dx

So, the total area between the curve and x-axis = sum of area of all strips = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Let f(x) > 0 be continuous on [a,b]. The area of the region between the graph of f and the x-axis is

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.66 Using an area to evaluate a definite integral 

Evaluate Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET 0 < a < b. 

Sol. We sketch the region under the curve y = x, a £ x £ b (figure) and see that it is a trapezoid with height (b -a) and bases a and b.

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

The value of the integral is the area of this trapezoid :

Thus =

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

and so on.

Notice that x2/2 is an antiderivative of x, further evidence of a connection between antiderivatives and summation.

(i) To find impulse

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET so implies = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Ex.67 If F = kt then find impulse at t = 3 sec.

so impulse will be area under f - t curve

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET                               

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET = Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

2. To calculate work done by force :

 Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

Resolution of Vectors: Motion in a Plane - Notes | Study Physics Class 11 - NEET

So area under f - x curve will give the value of work done.

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