Q.1 (d) A force 30 N is acting at point P(3, 2, 1) m. The force makes 35° & 85° with x & y axes respectively. Express the force in vector form.
Best answer
Direction cosines satisfy \(\cos^2\theta_x + \cos^2\theta_y + \cos^2\theta_z = 1\):
\(\cos^2\theta_z = 1 - \cos^2 35^\circ - \cos^2 85^\circ = 1 - 0.6710 - 0.0076 = 0.3214 \Rightarrow \cos\theta_z = 0.5669\), \(\theta_z = 55.46^\circ\) (taking the positive root)
\(F_x = 30\cos35^\circ = 24.57\) N, \(F_y = 30\cos85^\circ = 2.61\) N, \(F_z = 30 \times 0.5669 = 17.01\) N
Answer: \(\vec F = 24.57\,\hat i + 2.61\,\hat j + 17.01\,\hat k\) N, acting through P(3, 2, 1) m. (Check: \(\sqrt{24.57^2 + 2.61^2 + 17.01^2} = 30\) N ✔)
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