Question

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

Answer 1

The position vectors `bar"a", bar"b", bar"c"` and `bar"d"` of the points A, B, C and D w.r.t. the origin are

`bar"a" = - hat"i" + 2hat"j" + 3hat"k"`, 

`bar"b" = 3hat"i" - 2hat"j" + hat"k"`,

`bar"c" = 2 hat"i" + hat"j" + 3hat"k"`

And `bar"d" = -hat"i" - 2hat"j" + 4hat"k"`

∴ `bar"AB" = bar"b" - bar"a"`

= `(3hat"i" - 2hat"j" + hat"k") - (- hat"i" + 2hat"j" + 3hat"k")`

= `4hat"i" - 4hat"j" - 2hat"k"`

`bar"AC" = bar"c" - bar"a"`

= `(2 hat"i" + hat"j" + 3hat"k") - (- hat"i" + 2hat"j" + 3hat"k")`

= `3hat"i" - hat"j" + 0hat"k"`

`bar"AD" = bar"d" - bar"a"`

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

= `0hat"i" -  4hat"j" + hat"k"`

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

= `1/6|(4,-4,-2),(3,-1,0),(0,-4,1)|`

= `1/6[4(- 1 + 0) + 4(3 - 0) - 2(- 12 + 0)]`

= `1/6(- 4 + 12 + 24)` 

= `1/6 xx 32`

= `16/3` cubic units.