The unit vectors orthogonal to $3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ and coplanar with $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are

Question

The unit vectors orthogonal to $3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ and coplanar with $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are, $\pm \frac{1}{\sqrt{5}}(2 \hat{\mathbf{i}}-\hat{\mathbf{k}})$, $\pm \frac{1}{\sqrt{10}}(3 \hat{\mathbf{j}}-\hat{\mathbf{k}})$, $\pm \frac{1}{\sqrt{13}}(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}})$, $\pm \frac{1}{\sqrt{17}}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$

MARKS App · Accepted Answer

A vector which is coplanar to the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ is $\mathbf{a}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})+\mathbf{b}(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})$ $(2 a+b) \hat{\mathbf{i}}+(a-b) \hat{\mathbf{j}}+(a+b) \hat{\mathbf{k}}$ ...(i) Eq. (i) is orthogonal to $3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ $\begin{aligned} & 3(2 a+b)+2(a-b)+6(a+b) =0 \ & 14 a+7 b =0 \ & 2 a+b =0 \end{aligned}$ Required unit vector, $\begin{aligned} & = \pm \frac{(2 a+b) \hat{\mathbf{i}}+(a-b) \hat{\mathbf{j}}+(a+b) \hat{\mathbf{k}}}{\sqrt{(2 a+b)^2+(a-b)^2+(a+b)^2}} \ & = \pm \frac{3 a \hat{\mathbf{j}}-a \hat{\mathbf{k}}}{\sqrt{(3 a)^2+a^2}}= \pm \frac{3 \hat{\mathbf{j}}-\hat{\mathbf{k}}}{\sqrt{10}} \end{aligned}$ Hence, option (b) is correct.

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