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Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE PDF Download

1. Definitions:
A Vector may be described as a quantity having both magnitude & direction. A vector is generally represented by a directed line segment, say  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE . A is called the initial point & B is called the terminal point. The magnitude of vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is expressed byImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE.

Zero vector  a vector of zero magnitude i.e.which has the same initial & terminal point, is called a Zero vector. It is denoted by O.

Unit vector a vector of unit magnitude in direction of a vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is called unit vector along Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE and is denoted by aˆ symbolicallyImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Equal vectors  two vectors are said to be equal if they have the same magnitude, direction & represent the same physical quantity.  

Collinear vectors  two vectors are said to be collinear if their  directed line segments are parallel disregards to their direction. Collinear vectors are also called Parallel vectors. If they have the same direction they are named as like vectors otherwise.

unlike vectors. Symbolically,  two  non  zero  vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are collinear if and only if, 

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEwhere K ∈ R  Coplanar vectors a given number of vectors are called coplanar if their line segments are all parallel to the same plane. Note that  “two vectors are always coplanar”. position vector  let O be a fixed origin, then the position vector of a point P is the vector →OP . If  a & b & position vectors of two point A and B, then  , →AB = b a

− =  pv of B −  pv of A  .


2. Vector addition: 

If two vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are represented by → Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE , then their sum Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a vector represented by Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE , where OC is the diagonal  of the parallelogram OACB.

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE (commutative)

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  (associativity)

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


3. Multiplication of vector by scalars :

If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a vector & m is a scalar, then mImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a vector parallel toImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE whose modulus is |m| times that  of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE. This multiplication is called Scalar multiplication. If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEare vectors & m, n are scalars, then:

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


4. Section formula :

If  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are the position vectors of two points A & B then the p.v. of a point which divides AB in the ratio m : n is given by :  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE 

note p.v..of mid point of AB = Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


5. Direction Cosines 

Let  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE the angles which this  vector makes  with the +ve directions OX,OY & OZ are called Direction angles & their cosines are called the Direction cosinesImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Note that, cos²    αααα + cos² ββββ + cos² ΓΓΓΓ = 1.


6. Vector equation of a line:
Parametric vector  equation of a line passing through two point Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

 is given by,Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where t is a parameter. If the line passes  through  the  point Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE& is parallel to the vector then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  its equation is, Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Note that the equations of the bisectors of the angles between the lines Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


7. Test of collinearity  :
Three points A,B,C with position vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

 respectively are collinear,  if & only if there exist scalars x , y , z  not all zero simultaneously such that  ; Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEwhere x + y + z = 0.


8. Scalar product of two vectors:

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE note that if θ is acute then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE &  if θ is obtuse  then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE (commutative)

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Note: That vector component of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE and perpendicular to Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE the angle φ between Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is given by  cos φ = Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE 0 ≤ φ ≤ π.

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Note : (i) Maximum  value  of  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(ii) Minimum  values  of   Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


(iii) Any  vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEcan  be  written  as   Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(iv) A vector in the direction of the bisector of the angle between the two vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  Hence bisector  of  the  angle  between the two vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Bisector of the exterior angle between Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


9. Vector product of two vectors :

(i) If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are two vectors   &   θ   is   the  angle  between  them  then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is the unit vector perpendicular to both Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  such  that     Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE forms a  right  handed  screw  system.

(ii) Lagranges Identity : for any two vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(iii) Formulation of vector product in terms of scalar product:
The vector product Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is the vector  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE such that.

(i) Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE form a right handed system.

(iv) Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are parallel (collinear) Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where K is a scalar..

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(vi) Geometrically Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE of  the  parallelogram  whose  two  adjacent  sides are represented by Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(vii) Unit vector perpendicular to the plane of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


- A vector of magnitude ‘r ’ & perpendicular to the palne of  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

If θ is the angle between Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(viii) Vector area  If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are the pv’s of 3 points  A, B & C then the vector area of triangle ABC Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE The points A, B & C are collinear if  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Area of  any quadrilateral whose diagonal vectors are Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


10. Shortest distance between two lines:
If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & are also not parallel are called SKEW LINES. For Skew lines the direction of the shortest distance would be perpendicular to both the lines. The magnitude of the shortest distance vector would be equal to that of the projection of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE along the direction of the line of shortest distance, Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is parallel to  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE i. e .  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

1. The  two  lines  directed  along Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE will  intersect  only  if  shortest distance = 0  i.e. 

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE lies in the plane containing Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

2. If two lines are given by Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE i.e.  they are parallel then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


11. Scalar triple product / box product / mixed product :
The  scalar  triple  product  of  three  vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is  defined  as :

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE sin θ cos φ where θ is the angle between Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE it is also defined as Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE spelled as box product .Scalar triple product geometrically represents the volume of the parallelopiped  whose three  couterminous edges are represented by  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

In a scalar triple product the position of dot & cross can be interchanged i.e.Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are non coplanar vectors .

If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are coplanar  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Scalar product of three vectors, two of which are equal or parallel is 0 i.e.Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Note : If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are  non − coplanar  then  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE for  right  handed  system  & Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE for left handed system .

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

The volume of the tetrahedron OABC with O as origin & the pv’s of A, B and C being Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  respectively is given by Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

The positon vector of the centroid of a tetrahedron if the pv’s of its angular vertices areImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE are given by  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Note that this is also the point of concurrency of the lines joining the vertices to the centroids of the opposite faces and is also called the centre of the tetrahedron. In case the tetrahedron is regular it is equidistant from the vertices and the four faces of the tetrahedron .


12. Vector Triple Product : LetImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE be any three vectors, then the expression Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a vector & is called a vector triple product .

Geometrical interpretation of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Consider the expression Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE which itself is a vector, since it is a cross  product  of  two  vectors  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Now Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is  a vector perpendicular to the plane containing  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE butImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a vector perpendicular to the plane  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE therefore Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is  a vector lies in the plane of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE and perpendicular to Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Hence we can express  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE in terms of  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE i.e. Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEwhere x & y are scalars .

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


13. Linear combinations / Linearly Independence and Dependence of Vectors :
Given a finite set of vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE then the vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE 

is called a linear combination of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE for any x, y, z ...... ∈ R. We have the following results :

(a) Fundamental theorem in plane :  Let Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE be  non zero ,  non collinear vectors . Then any vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE coplanar with Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE can be expressed uniquely as a linear combination of  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE There exist some unique x,y ∈ R such that Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(b) Fundamental theorem in space : Let Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE be non−zero, non−coplanar vectors in space. Then  any vector Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE can be uniquily expressed as a linear combination of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

There exist some unique x,y ∈ R such that Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(c) If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are  n  non zero vectors,  & k1, k2, .....kn  are n  scalars & if the linear  combination Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Linearly independent vectors

(d) If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  are  not Linearly independent  then  they  are said  to be Linearly dependent vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE &  if  there  exists at least  one kr ≠ 0 then 

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEare said to be linearly dependent .

Note:

If Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is expressed as a linear combination of vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE form a linearly dependent set of vectors. In general , every set of four vectors is a linearly dependent system. Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are Linearly independent set of vectors. For Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Two vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are linearly dependent ⇒ Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is parallel toImportant Formulas: Vector | Mathematics (Maths) Class 12 - JEE linear dependence of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE Conversely if Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE then Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are linearly independent  .

If  three vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are linearly dependent, then they are coplanar i.e.  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

conversely, if Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE  then the vectors are linearly independent.


14. Coplanarity of  vectors:
Four points A, B, C, D with position vectors Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where,  x + y + z + w = 0.


15. Reciprocal system of vectors:
If  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are two sets of non coplanar vectors such that Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

then the two systems are called Reciprocal System of vectors. 

Note: Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE


16. Equation of a plane 
(a) The  equation Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE represents  a  plane  containing  the  point  with

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a vector normal to the plane .Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is the general equation of a plane.

(b) Angle between the 2 planes is the angle between 2 normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane.


17. Application of vectors:
(a) Work done against a constant force Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE over a displacement Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is defined as

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(b) The  tangential  velocity Important Formulas: Vector | Mathematics (Maths) Class 12 - JEEof  a  body moving  in a circle is given by  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is the pv of the point P.

(c) The moment of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE about ’O’ is defined as Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is the pv  of  P  wrt  ’O’.  The direction  of Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is  along  the  normal  to  the plane OPN   such   that  Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE 

 form  a right  handed system.

Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE

(d) Moment of the couple = Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE where Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE are  pv’s of the point of the application of the forces Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE.

The document Important Formulas: Vector | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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