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Mathematics Question on Vector Algebra

Find the volume of the tetrahedron whose vertices are A(3,7,4)A(3,7,4), B(5,−2,3)B(5, - 2, 3), C(−4,5,6)C(- 4, 5, 6) and D(1,2,3)D(1, 2,3).

A

1212 cu. units

B

233\frac{23}{3} cu. units

C

1515 cu. units

D

463\frac{46}{3} cu. units

Answer

463\frac{46}{3} cu. units

Explanation

Solution

We have, AB→=2i^−9j^−k^\overrightarrow{AB} =2\hat{i}-9\hat{j}-\hat{k}, AC→=−7i^−2j^+2k^\overrightarrow{AC} =-7\hat{i}-2\hat{j}+2\hat{k}, AD→=−2i^−5j^−k^\overrightarrow{AD} =-2\hat{i}-5\hat{j}-\hat{k} ∴\therefore Volume of tetrahedron =16[AB→ AC→ AD→]= \frac{1}{6}\left[\overrightarrow{AB}\,\overrightarrow{AC}\,\overrightarrow{AD}\right] =16∣2−9−1 −7−22 −2−5−1∣= \frac{1}{6}\begin{vmatrix}2&-9&-1\\\ -7&-2&2\\\ -2&-5&-1\end{vmatrix} =16[2(2+10)+9(7+4)−1(35−4)]= \frac{1}{6}\left[2\left(2 +10\right) + 9\left(7 + 4\right) -1\left(35 - 4\right)\right] =16[24+99−31]= \frac{1}{6}\left[24 + 99 - 31\right] =926=463= \frac{92}{6} = \frac{46}{3} cu. units