Question

The volume of tetrahedron whose vertices are

A = (3, 2, 1) ,~B = (1, 2, 4),~ C = (4, 0, 3),~ D = (1, 1, 7)~will be $\underset{–}{}$cubic units

• A. 5
• B.
• C.
• D. None of the above

Answer 1

The correct option is B

$Givenverticesare\overrightarrow{A}=(3,2,1)\overrightarrow{B}=(1,2,4)\overrightarrow{C}=(4,0,3)\overrightarrow{D}=(1,1,7)Sothethreecoterminousedgeswecanformare\overrightarrow{AB},\overrightarrow{AC}and\overrightarrow{AD}Right?So\overrightarrow{AB}=\left(\right(1-3),(2-2),(4-1\left)\right)=(-2,0,3)So\overrightarrow{AC}=\left(\right(4-3),(0-2),(3-1\left)\right)=(1,-2,2)So\overrightarrow{AD}=\left(\right(1-3),(1-2),(7-1\left)\right)=(-2,-1,6)Sovolume=\frac{1}{6}\left|\right[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}\left]\right|Rememberthatwehavetotaketheabsolutevalueofscalartripleproduct.So\left[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}\right]=\left|\begin{array}{ccc}-2& 0& 3\\ 1& -2& 2\\ -2& -1& 6\end{array}\right|=20-15=5Sovolumeoftetrahedron=\frac{1}{6}\left[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}\right]=\frac{1}{6}\times 5=\frac{5}{6}cubicunits.$