Question

With respect to rectangular coordinate system $OXYZ$, $\overrightarrow{a}$ has components $(2,4,8)$ . The system is rotated through an angle of $\frac{\pi }{2}$ about X-axis. The components of $\overrightarrow{a}$ in the new system are

• A. $(2,8,4)$
• B. $(2,8,-4)$
• C. $(2,-8,4)$
• D. $(2,-8,-4)$

Answer 1

Answer: (C)

$(2,-8,4)$

$\overrightarrow{a}=(2,4,8)$

Compnents of $\overrightarrow{a}$ are ${\overrightarrow{a}}_1,{\overrightarrow{a}}_2,{\overrightarrow{a}}_3$ along coordinates axes

$a_1=2,a_2=4,a_3=8$,$a_2=a\cos \alpha ,a_3=a\sin \alpha$

let new components of $\overrightarrow{a}$ be $A_1,A_2,A_3$

$A_1$ remains the same as system is rotated along X axis.

$A_2=a\cos (\alpha +\frac{\pi }{2})=-a\sin \alpha =-a_3=-8$

$A_3=a\sin (\alpha +\frac{\pi }{2})=a\cos \alpha =a_2=4$