Question

The volume of tetrahedron whose vertices are A(3, 7, 4), B(5, -2, 3), C(-4, 5, 6), D(1, 2, 3) is ______.

Options

• \[\frac{43}{6}\] cu.units

• 43 cu.units

• \[\frac{46}{3}\] cu.units

• \[\frac{6}{43}\] cu.units

Answer 1

The volume of tetrahedron whose vertices are A(3, 7, 4), B(5, -2, 3), C(-4, 5, 6), D(1, 2, 3) is \[\frac{46}{3}\] cu.units.

Explanation:

Volume of tetrahedron = `1/6 [bar"AB"  bar"AC"  bar"AD"]`

`bar"AB" = 2hat"i" - 9 hat"j" - hat"k"`,

`bar"AC" = - 7hat"i" - 2hat"j" + 2hat"k"`,

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

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

=2(2 + 10) + 9(7 + 4) - 1 (35 - 4) = 92

∴ Volume of tetrahedron = `1/6(92)`

`= 46/3` cubic units