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.

Options

• 91

• 94

• 92

• 93

Answer 1

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 92 cubic units.

Explanation:

We have

AB = `(5 - 3)hat"i" + (- 2 - 7)hat"j" + (3 - 4)hat"k"`

= `2hat"i" - 9hat"j" - hat"k"`

AC = `(- 4 - 3)hat"i" + (5 - 7)hat"j" + (6 - 4)hat"k"`

`= - 7hat"i" - 2hat"j" + 2hat"k"`

and AD = `(1 - 3)hat"i" + (2 - 7)hat"j" + (3 - 4)hat"k"`

`= - 2hat"i" - 5hat"j" - hat"k"`

∴ Volume of the parallelopiped with AB, AC and AD on the co-terminus edges

= |[AB AC AD]| = `|(2,-9,-1),(-7,-2,2),(-2,-5,-1)|`

= |2(2 + 10) + 9(7 + 4) - 1(35 - 4)|

= |2 (12) + 9(11) - 1(31)|

= |(24 + 99 - 31)|

= |92|

= 92 cubic units