A unit vector coplanar with $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}+\hat{j}+\hat{k}$ and perpendicular to…

A unit vector coplanar with $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}+\hat{j}+\hat{k}$ and perpendicular to $\hat{i}+\hat{j}-\hat{k}$ is
  1. $+\frac{1}{\sqrt{2}}(-\hat{j}-\hat{k})$
  2. $\frac{(\hat{\mathrm{j}}-\hat{\mathrm{k}})}{\sqrt{2}}$
  3. $\frac{-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}}{\sqrt{5}}$
  4. $+\frac{1}{\sqrt{26}}(\hat{\mathrm{j}}+5 \hat{\mathrm{k}})$

Solution

Let $\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{c}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$. Then, required unit vectors are given by $\bar{\alpha}= \pm \frac{\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \cdot}{|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|}$
Now, $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}$ $\begin{array}{ll} \therefore & =2(\hat{i}+\hat{j}+\hat{k})-1(2 \hat{i}+\hat{j}+\hat{k})=\hat{j}+\hat{k} \\ \therefore & |\bar{a} \times(\bar{b} \times \bar{c})|=\sqrt{1+1}=\sqrt{2} \end{array}$ Hence, required unit vectors are $\bar{\alpha}= \pm \frac{\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{2}}$

Asked in: MHT CET 2024 (09 May Shift 2)

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