Question

If the vectors $2\hat{i}-3\hat{j},\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}-\hat{k}$ form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is

• A. $8$
• B. $10$
• C. $4$
• D. $14$

Answer 1

Answer: (C)

$4$

Let
$OA=2\hat{i}-3\hat{j}=\overline{a}$
$OB=\hat{i}+\hat{j}-\hat{k}=\overline{b}$
$OC=3\hat{i}-\hat{k}=\overline{c}$

Hence volume of parallelepiped is
$\overline{a}.(\overline{b}\times \overline{c})$
$=(2\hat{i}-3\hat{j}).\left[\right(\hat{i}+\hat{j}-\hat{k})\times (3\hat{i}-\hat{k}\left)\right]$
$=(2\hat{i}-3\hat{j}).[-\hat{i}-2\hat{j}-3\hat{k}]$
$=(2\hat{i}-3\hat{j}).(-\hat{i}-2\hat{j}-3\hat{k})$
$=-2+6$
$=4$
Hence volume is $4$.