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4. LAWS OF ADDITION AND SUBTRACTION OF VECTORS
4.1 Triangle rule of addition: Steps for addicting two vector representing same physical quantity by triangle law.
(i) Keep vectors s.t. tail of one vector coincides with head of others.
(ii) Join tail of first to head of the other by a line with arrow at head of the second.
(iii) This new vector is the sum of two vectors. (also called resultants)
(i) (ii) (iii)
Take example here.
Q. A boy moves 4 m south and then 5 m in direction 37° E of N. Find resultant displacement.
4.2 Polygon Law of addition :
This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors s.t. tail of next vector lies on head of former.
When we connect the tail of first vector to head off last we get resultant of all the vectors.
4.3 Parallelogram law of addition :
(i) Keep two vectors such that their tails coincide.
(ii) Draw parallel vectors to both of them considering both of them as sides of a parallelogram.
(iii) Then the diagonal drawn from the point where tails coincide represents the sum of two vectors, with its tail at point of coincidence of the two vectors.
(i) (ii) (iii)
Note: and thus [Commutative Law]
Note: Angle between 2 vectors is the angle between their positive directions.
Suppose angle between these two vectors is θ, and
(AD)2 = (AE)2 +(DE)2
= (AB + BE)2 + (DE)2
= (a +b cosθ)2 + (b sinθ)2
= a2 + b2 cos2θ + 2ab cosθ + b2 sin2θ
= a2 + b2 + 2ab cosθ
Thus, AD =
angle α with vector a is
tan α = =
Important points :
i.e., θ = 0º, i.e. vectors are like or parallel and Rmax = A + B.
5. SUBTRACTION OF VECTORS :
Negative of a vector say is a vector of the same magnitude as vector but pointing in a direction opposite to that of .
Thus, can be written as or is really the vector addition of and .
Suppose angle between two vectors and is θ. Then angle between and will be 180° -θ as shown in figure.
Magnitude of will be thus given by
S = =
or S = ...(i)
For direction of we will either calculate angle α or β, where,
tanα = = ...(ii)
or tanβ = = ...(iii)
Ex.3 Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude.
⇒ Sol. =
⇒ α = 30°
[Here shows what is angle between both vectors = 120° and not 60°]
Note: or can also be found by making triangles as shown in the figure. (a) and (b)
Ex.4 Two vectors of equal magnitude 2 are at an angle of 60° to each other find magnitude of their sum & difference.
Ex.5 Find and in the diagram shown in figure. Given A = 4 units and B = 3 units.
Sol. Addition :
= = units
tanα = = = 0.472
a = tan-1(0.472) = 25.3°
Thus, resultant of and is units at angle 25.3° from in the direction shown in figure.
Subtraction : S =
and tanθ =
= = 1.04
∴ α = tan-1 (1.04) = 46.1°
Thus, is units at 46.1° from in the direction shown in figure.
6. Unit Vector and Zero vector
Unit vector is a vector which has a unit magnitude and points in a particular direction. Any vector can be written as the product of unit vector in that direction and magnitude of the given vector.
A unit vector has no dimensions and unit. Unit vectors along the positive x-, y-and z-axes of a rectangular coordinate system are denoted by and respectively such that
A vector of zero magnitudes is called zero or a null vector. Its direction is arbitrary.
Ex.6 A unit vector along East is defined as . A force of 105 dynes acts west wards. Represent the force in terms of .