Physics Complete: Vector Resolution

A vector will be generally having components in different directions like x,y and z. Dividing the vector into components along this directions is called resolution of vectors.

We are able to identify value of vectors along the corresponding directions with the help of components 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. 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 = OQ + QP

But since OQ is parallel to a, and QP is parallel to b, we can write :

OQ = λ a, and QP = µ b where λ and µ are real numbers.

Therefore, A = λ a + µ b.

We say that A has been resolved into two component vectors λ a and μ b along a and b 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 a rectangular coordinate system using vectors of unit magnitude.

Unit vectors: A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only. Unit vectors along the x-, y and z-axes of a rectangular coordinate system are denoted by iˆ , jˆ and kˆ , respectively, as shown in Figure below.

These unit vectors are perpendicular to each other.

If we multiply a unit vector, say n by a scalar, the result is a vector λ = λ n . In general, a vector A can be written as A = |A|n.

Vector resolution in two dimensions basing on Unit vectors :

Consider a vector A that lies in x-y plane as shown in Figure below. We draw lines from the head of A perpendicular to the coordinate axes and get vectors A1 and A2 such that A1 + A2 = A. Since A1 is parallel to I and A2 is parallel to J , we have :

A1 = Ax i, A2 = Ay j where Ax and Ay are real numbers.

So we can represent the vector as shown.

A = Ax i+ Ay j

Using simple trigonometry, we can express Ax and Ay in terms of the magnitude of A and the angle θ it makes with the x-axis :

Ax = A cos θ
Ay = 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 Ax and Ay If A and θ are given, Ax and Ay can be obtained using Eq. If Ax and Ay are given, A and θ. Then we can deduce the following relations .