The unit vector which is orthogonal to the vector $5 \hat{i}+2 \hat{j}+6 \hat{k}$ and is coplanar with the…

The unit vector which is orthogonal to the vector $5 \hat{i}+2 \hat{j}+6 \hat{k}$ and is coplanar with the vectors $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ is
  1. $\frac{2 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{41}}$
  2. $\frac{2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}}{\sqrt{29}}$
  3. $\frac{-3 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{10}}$
  4. $\frac{2 \hat{i}-8 \hat{j}+\hat{k}}{69}$

Solution

Let $\overline{\mathrm{a}}=5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\bar{c}=\hat{i}-\hat{j}+\hat{k}$
Then, required unit vectors are given by $\bar{\alpha}= \pm \frac{\bar{a} \times(\bar{b} \times \bar{c})}{|\overline{\mathrm{a}} \times(\bar{b} \times \bar{c})|}$
Now, $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}$ $\begin{aligned} & =9(2 \hat{i}+\hat{j}+\hat{k})-18(\hat{i}-\hat{j}+\hat{k}) \\ & =27 \hat{j}-9 \hat{k} \end{aligned}$ $\therefore \quad|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|=\sqrt{729+81}=\sqrt{810}=9 \sqrt{10}$ Hence, required unit vectors are $\bar{\alpha}= \pm \frac{27 \hat{\mathrm{j}}-9 \hat{\mathrm{k}}}{9 \sqrt{10}}= \pm \frac{3 \hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{10}}$

Asked in: MHT CET 2024 (02 May Shift 2)

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