ˆ i = ˆ j = ˆ k = ( . ) These unit vectors are perpendicular to each other. In this text, they are printed in bold face with a cap (^) to distinguish them from other vectors. Since we are dealing with motion in two dimensions in this chapter, we require use of only two unit vectors.
If we multiply a unit vector, say ˆ n by a scalar, the result is a vector λλλλλ = λ ˆ n . In general, a vector A can be written as A = | A | ˆ n ( . ) where ˆ n is a unit vector along A . We can now resolve a vector A in terms of component vectors that lie along unit vectors i ˆ and ɵ j .
Consider a vector A that lies in x-y plane as shown in Fig. . (b). We draw lines from the head of A perpendicular to the coordinate axes as in Fig.
. (b), and get vectors A and A such that A + A = A . Since A is parallel to ɵ i and A is parallel to ɵ j , we have : A = A x ɵ i , A = A y ɵ j ( . ) where A x and A y are real numbers.
Thus, A = A x ɵ i + A y ɵ j ( . ) This is represented in Fig. . (c).
The quantities A x and A y are called x- , and y- components of the vector A . Note that A x is itself not a vector, but A x ɵ i is a vector, and so is A y ɵ j . Using simple trigonometry, we can express A x and A y in terms of the magnitude of A and the angle θ it makes with the x -axis : A x = A cos θ A y = A sin θ ( . ) As is clear from Eq.
( . ), a component of a vector can be positive, negative or zero depending on the value of θ . Now, we have two ways to specify a vector A in a plane. It can be specified by : (i) its magnitude A and the direction θ it makes with the x -axis; or (ii) its components A x and A y If A and θ are given, A x and A y can be obtained using Eq.
( . ). If A x and A y are given, A and θ can be obtained as follows : cos = A ( . ) And , ( .
) Let R be their sum. We have R = A + B