**RESOLUTION OF VECTORS**

Resolution of vectors is the opposite action of addition of vectors. For a given vector A, we may find a pair of vectors A-; and A; in any two given direction. Refer Fig. 2.5(a).

Fig. 2.5(b) shows that vector A is replaced by its components A1 and A2 and A is no longer operative.

Here, A_{1} and A_{2} are called the component vectors. The stipulated directions may include any angle e. If e = 90°, the components are called rectangular components.

We can also resolute A in space into three orthogonal components which are not in the same plane.

We first resolve A into two components A3 and A4 as shown in Fig. 2.6. Then A_{4} can be resolved into A_{1} and A_{2} which are in x and y directions respectively. Thus, finally A is resolved into three orthogonal components A_{1}, A_{2}, A_{3} in x, y and z directions.

Let

θ = Angle made by A with x axis

θ = Angle made by A with y axis

θ = Angle made by A with z axis.

If l, m and n are direction cosines in x, y, and z directions respectively, we can write

L =cos θ_{x } A_{1}=Al

M=cos θ _{y}, A_{2} =Am

N =cos θ _{z}, A3 =An

From (1.2) and (1.3) à A^{2} =A_{1}^{2} +A^{2}_{2}+A_{3}^{2}

A^{2} =A^{2} (L^{2}+M^{2}=N^{2} ) L^{2} +M^{2} +N^{2} =1