Question

If $ABC$ and $D$ are $(3,7,4),(5,-2,-3),(-4,5,6)$ and $(1,2,3)$ respectively, then the volume of the parallelopiped with $AB,AC$ and $AD$ as the co-terminus edges, is.......cubic units.

• A. $91$
• B. $94$
• C. $92$
• D. $93$

Answer 1

Answer: (C)

We have
$AB=(5-3)\hat{i}+(-2-7)\hat{j}+(3-4)\hat{k}$
$=2\hat{i}-9j-\hat{k}$
$AC=(-4-3)\hat{i}+(5-7)\hat{j}+(6-4)\hat{k}$
$=-7\hat{i}-2\hat{j}+2\hat{k}$
and $AD=(1-3)\hat{i}+(2-7)\hat{j}+(3-4)\hat{k}$
$=-\widehat{2i}-5\hat{j}-\hat{k}$
$\therefore$ Volume of the parallelopiped with $AB,AC$ and $AD$ on the co-terminus edges
$=|\left[ABACAD\right]|=\left|\begin{array}{ccc}2& -9& -1\\ -7& -2& 2\\ -2& -5& -1\end{array}\right|$
$=|2(2+10)+9(7+4)-1(35-4)|$
$=|2(12)+9(11)-1(31)|$
$=|(24+99-31)|$
$=|92|=92$ cubic units