Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the intersection face diagonals of a cube of side x in plane XOY and YOZ, respectively, with respect to reference frame at the point of intersection of the vectors and sides of cube as the axes, the components of vector $\overrightarrow{r} = \overrightarrow{a} \times \overrightarrow{b}$ are

• A. x, - x, x
• B. –x², -x², x²
• C. x², - x², x²
• D. x, x², - x.

Answer 1

Answer: (C)

x², - x², x²

Answer 2

Here $\overrightarrow{a} = x\widehat{i} + x\widehat{j}$ and $\overrightarrow{b} = x\widehat{j} + x\widehat{k}$ since $\overrightarrow{R}$

= $\overrightarrow{a} \times \overrightarrow{b}$ we get $\overrightarrow{R}$= $\left| \begin{matrix} \widehat{i} & \widehat{j} & \widehat{k} \\ x & x & 0 \\ 0 & x & x \end{matrix} \right|$ = x²$\widehat{i}$ - x²$\widehat{j}$ + x²$\widehat{k}$

Clearly, the components are R_x = x², R_y = -x², R_z = x²