⊳ Fig. . (a) Two vectors A and B with their tails brought to a common origin. (b) The sum A + B obtained using the parallelogram method.
(c) The parallelogram method of vector addition is equivalent to the triangle method. . RESOLUTION OF VECTORS Let a and b be any two non-zero vectors in a plane with different directions and let A be another vector in the same plane (Fig. .
). A can be expressed as a sum of two vectors — one obtained by multiplying a by a real number and the other obtained by multiplying b by another real number. To see this, let O and P be the tail and head of the vector A . Then, through O, draw a straight line parallel to a , and through P, a straight line parallel to b .
Let them intersect at Q . Then, we have A = OP = O Q + Q P ( . ) But since O Q is parallel to a , and Q P is parallel to b , we can write : O Q = λ a , and Q P = µ b ( . ) where λ and µ are real numbers.
Therefore, A = λ a + µ b ( . ) Fig. . (a) Two non-colinear vectors a and b .
(b) Resolving a vector A in terms of vectors a and b . We say that A has been resolved into two component vectors λ a and µ b along a and b Fig. . (a) Unit vectors ɵ i , ɵ j and ɵ k lie along the x-, y-, and z-axes.
(b) A vector A is resolved into its components A x and A y along x-, and y- axes. (c) A and A expressed in terms of ɵ i and ɵ j . respectively. Using this method one can resolve a given vector into two component vectors along a set of two vectors – all the three lie in the same plane.
It is convenient to resolve a general vector along the axes of