Vector Algebra Question 178
Question: The volume of the tetrahedron, whose vertices are given by the vectors $ -\mathbf{i}+\mathbf{j}+\mathbf{k},\mathbf{i}-\mathbf{j}+\mathbf{k} $ and $ \mathbf{i}+\mathbf{j}-\mathbf{k} $ with reference to the fourth vertex as origin, is
Options:
A) $ \frac{5}{3} $ cubic unit
B) $ \frac{2}{3} $ cubic unit
C) $ \frac{3}{5} $ cubic unit
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
- Volume of tetrahedron $ ABCD $ is, $ \frac{1}{6}|\overrightarrow{AB}\times \overrightarrow{AC},.,\overrightarrow{AD}| $ , where $ A(-1,,1,,1), $ $ B(1,,-1,,1), $ $ C(1,,1,,-1) $ and $ D(0,,0,,0). $
$ =\frac{1}{6}|(2\mathbf{i}-2\mathbf{j})\times (2\mathbf{i}-2\mathbf{k}),.,(\mathbf{i}-\mathbf{j}-\mathbf{k})| $
$ =\frac{1}{6}| ,\begin{matrix} 2 & -2 & 0 \\ 2 & 0 & -2 \\ 1 & -1 & -1 \\ \end{matrix}, |=\frac{1}{6}(-4)=-\frac{2}{3}=\frac{2}{3} $ cubic unit.
BETA
