MHT CET · Maths · Vector Algebra
If \(\bar{u}=\hat{\imath}-2 \hat{\jmath}+\hat{k}, \bar{v}=3 \hat{\imath}+\hat{k}\) and \(\bar{w}=\hat{\jmath}-\hat{k}\), then the volume of the parallelopiped
with \(\bar{u} \times \bar{v}, \bar{u}+\bar{w}\) and \(\bar{v}+\bar{w}\) as coterminus edges is
- A 12 cubic units
- B 10 cubic units
- C 24 cubic units
- D 20 cubic units
Answer & Solution
Correct Answer
(C) 24 cubic units
Step-by-step Solution
Detailed explanation
\(\bar{u} \times \bar{v}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 3 & 0 & 1\end{array}\right|=\hat{i}(-2)-\hat{j}(1-3)+\hat{k}\) \((0+6)=-2 \hat{i}+2 \hat{j}+6 \hat{k}\)
\(\hat{u}+\hat{w}=\hat{i}-\hat{j}\)
\(\hat{v}+\hat{w}=3 \hat{i}+\hat{j}\)
\(\begin{aligned} \text { Volume of parallelopiped} &=(\bar{u} \times \bar{v}) \cdot[(\bar{u}+\bar{w}) \times(\bar{v}+\bar{w})]\end{aligned}\)
\(=\left|\begin{array}{ccc}-2 & 2 & 6 \\ 1 & -1 & 0 \\ 3 & 0 & 1\end{array}\right|=|-2(-1)-2(1)+6(-3)|=18\)
\(\hat{u}+\hat{w}=\hat{i}-\hat{j}\)
\(\hat{v}+\hat{w}=3 \hat{i}+\hat{j}\)
\(\begin{aligned} \text { Volume of parallelopiped} &=(\bar{u} \times \bar{v}) \cdot[(\bar{u}+\bar{w}) \times(\bar{v}+\bar{w})]\end{aligned}\)
\(=\left|\begin{array}{ccc}-2 & 2 & 6 \\ 1 & -1 & 0 \\ 3 & 0 & 1\end{array}\right|=|-2(-1)-2(1)+6(-3)|=18\)
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