Question

If $\overrightarrow{a}.\overrightarrow{c}=3$$,$ $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k},\overrightarrow{b}=\overrightarrow{j}-\overrightarrow{k},\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b,}$ Find $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$.

Answer 1

Explanation of the above question:

Find $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$:

Given,

$\overrightarrow{a}.\overrightarrow{c}=3$

$\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$

$\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}$

$\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ $=-[\overrightarrow{a}\overrightarrow{c}\overrightarrow{b}]$

$=-(\left(\overrightarrow{a}\times \overrightarrow{c}\right).\overrightarrow{b})$ By formulas of the scalar triple product

$=$$-(\overrightarrow{b}.\overrightarrow{b})$ $\left[\because \overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\right]$

$=-(\left(\overrightarrow{j}-\overrightarrow{k}\right).\left(\overrightarrow{j}-\overrightarrow{k}\right))$

$=-\left(2\right)$

Hence, the value of $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ is $-2$.