Question

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