Question

For any non zero, non collinear vectors $\overline{p}$ and $\overline{q}$ the value of
$\left[\overline{i}\overline{p}\overline{q}\right]\overline{i}+\left[\overline{j}\overline{p}\overline{q}\right]\overline{j}+\left[\overline{k}\overline{p}\overline{q}\right]\overline{k}$ is ___________

• A. $\overline{0}$
• B. $2(\overline{p}\times \overline{q})$
• C. $\overline{q}\times \overline{p}$
• D. $\overline{p}\times \overline{q}$

Answer 1

The correct option is D $\overline{p}\times \overline{q}$

Answer: D. $(\overline{p}\times \overline{q})$

Solution: We know that for any two non-collinear vectors $\overline{p}$ and $\overline{q}$, their cross product is non-zero

Also let $(\overline{p}\times \overline{q})$ =$\overline{a}$

Let $a_x$ denote the projection of $\overline{a}$ along $\overline{i}$ . Then,

$a_x$ = $\overline{a}$.$\overline{i}$

Similarily,

$a_y$ = $\overline{a}$.$\overline{j}$

and,

$a_z$ = $\overline{a}$.$\overline{k}$

So, the given scalar triple product can be decomposed as:

$\left[\overline{i}\overline{p}\overline{q}\right]\overline{i}+\left[\overline{j}\overline{p}\overline{q}\right]\overline{j}+\left[\overline{k}\overline{p}\overline{q}\right]\overline{k}$

= $\overline{i}$. $(\overline{p}\times \overline{q})$ $\overline{i}$ + $\overline{j}$. $(\overline{p}\times \overline{q})$ $\overline{j}$ + $\overline{k}$. $(\overline{p}\times \overline{q})$ $\overline{k}$

= $(\overline{i}$. $\overline{a})$ $\overline{i}$ + $(\overline{j}$. $\overline{a})$ $\overline{j}$ + $(\overline{k}$. $\overline{a})$ $\overline{k}$

= $a_x$ $\overline{i}$ + $a_y$ $\overline{j}$ + $a_z$ $\overline{k}$

= $\overline{a}$

= $\overline{p}\times \overline{q}$