Let $\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}$ and $\bar{w}=\hat{i}+3 \hat{k}$. If $\bar{u}$ is a unit vector,…

Let $\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}$ and $\bar{w}=\hat{i}+3 \hat{k}$. If $\bar{u}$ is a unit vector, then the maximum value of the scalar triple product $[\overline{\mathrm{u}} \overline{\mathrm{v}} \overline{\mathrm{W}}]$ is
  1. $\sqrt{6}$
  2. $\sqrt{10}$
  3. $\sqrt{13}$
  4. $\sqrt{89}$

Solution

$\begin{aligned} & \overline{\mathrm{v}} \times \overline{\mathrm{w}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & 2 & -1 \\ 1 & 0 & 3\end{array}\right|=\hat{\mathrm{i}}(6)-\hat{\mathrm{j}}(7)+\hat{\mathrm{k}}(-2)=6 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\ & \therefore|\overline{\mathrm{v}} \times \overline{\mathrm{w}}|=\sqrt{(6)^2+(7)^2+(2)^2}=\sqrt{89} \\ & \text { Also }|\overline{\mathrm{u}}|=1 \\ & \therefore[\overline{\mathrm{u}} \overline{\mathrm{v}} \overline{\mathrm{w}}]=\sqrt{89}\end{aligned}$

Asked in: MHT CET 2021 (23 Sep Shift 2)

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