# 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). - Mathematics and Statistics

Sum

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).

#### Solution

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.

Concept: Scalar Triple Product of Vectors
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