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Resolution of Vectors

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, A1 and A2 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 A4 can be resolved into A1 and A2 which are in x and y directions respectively. Thus, finally A is resolved into three orthogonal components A1, A2, A3 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 θx                            A1=Al

M=cos θ y,                   A2 =Am

N =cos θ z,                   A3 =An

From (1.2) and (1.3) à A2 =A12 +A22+A32

A2 =A2 (L2+M2=N2 )   L2 +M2 +N2 =1