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Chapter 1

Q. 1.7

Referring to Fig. (a), determine (1) the rectangular representation of the position vector A; and (2) the angles between A and each of the positive coordinate axes.

Referring to Fig. (a), determine (1) the rectangular representation of the position vector A; and (2) the angles between A and each of the positive coordinate axes.

Step-by-Step

Verified Solution

Part 1
We first resolve A into two components as shown in Fig. (b): A_z along the z-axis and A_{xy} in the xy-plane. (Once again we see that a carefully drawn sketch is an essential aid in performing vector resolution.) Because A, A_{z}, and A_{xy} lie in the same plane (a diagonal plane of the parallelepiped), we obtain by trigonometry

A_{z}=A\cos30^\circ=12\cos30^\circ=10.392\: m

A_{xy}=A\sin30^\circ=12\sin30^\circ=6\: m

The next step, illustrated in Fig. (c), is to resolve A_{xy} into the components along the coordinate axes:

A_{x}=A_{xy}\cos40^\circ=6\cos40^\circ=4.596\: m

A_{y}=A_{xy}\sin40^\circ=6\sin40^\circ=3.857\: m

Therefore, the rectangular representation of A is A=A_{x}i+A_{y}j+A_{z}k=4.60i+3.86j+10.39k\: m

Part 2
The angles between A and the coordinate axes can be computed from Eqs.(1.6):

A_x = A\cos θ_x \:       A_y = A\cosθ_y  \:          A_z = A\cosθ_z                 (1.6)

θ_{x}=\cos^{−1}\frac{A_{x}}{A}=\cos^{−1}\frac{4.596}{12}=67.5^\circ

 

θ_{y}=\cos^{−1}\frac{A_{y}}{A}=\cos^{−1}\frac{3.857}{12}=71.3^\circ

 

θ_{z}=\cos^{−1}\frac{A_{z}}{A} =\cos^{−1}\frac{10.392}{12}=30.0^\circ

These angles are shown in Fig. (d). Note that it was not necessary to compute θ_{z}, because it was already given in  Fig. (a).

Referring to Fig. (a), determine (1) the rectangular representation of the position vector A; and (2) the angles between A and each of the positive coordinate axes.
Referring to Fig. (a), determine (1) the rectangular representation of the position vector A; and (2) the angles between A and each of the positive coordinate axes.
Referring to Fig. (a), determine (1) the rectangular representation of the position vector A; and (2) the angles between A and each of the positive coordinate axes.