If $\bar{u}=\hat{\imath}-2 \hat{\jmath}+\hat{k}, \bar{v}=3 \hat{\imath}+\hat{k}$ and…

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
  1. 12 cubic units
  2. 10 cubic units
  3. 24 cubic units
  4. 20 cubic units

Solution

$\bar{u} \times \bar{v}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 3 & 0 & 1\end{array}\right|=\hat{i}(-2)-\hat{j}(1-3)+\hat{k}(0+6)=-2 \hat{i}+2 \hat{j}+6 \hat{k}$ $\hat{u}+\hat{w}=\hat{i}-\hat{j}$ $\hat{v}+\hat{w}=3 \hat{i}+\hat{j}$ $\begin{aligned} \text { Volume of parallelopiped } &=(\bar{u} \times \bar{v}) \cdot[(\bar{u}+\bar{w}) \times(\bar{v}+\bar{w})] \\ &=\left|\begin{array}{ccc}-2 & 2 & 6 \\ 1 & -1 & 0 \\ 3 & 0 & 1\end{array}\right|=|-2(-1)-2(1)+6(-3)|=18 \end{aligned}$

Asked in: MHT CET 2020 (12 Oct Shift 2)

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