Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and angle…

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and angle $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\overrightarrow{\mathrm{C}}$. Then $\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\frac{\sqrt{2}}{3} \hat{k}\right)$ is :
  1. $\frac{\sqrt{2}}{3} \hat{i}-\frac{1}{2} \hat{k}$
  2. $\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right) \hat{i}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3 \sqrt{2}}\right) \hat{j}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right) \hat{k}$
  3. $\frac{\sqrt{2}}{3} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\frac{1}{2} \hat{k}$
  4. $-\frac{\sqrt{2}}{3} \hat{i}+\frac{\sqrt{2}}{3} \hat{j}+\left(\frac{1}{2}+\frac{2 \sqrt{2}}{3}\right) \hat{k}$

Solution

$\overrightarrow{\mathrm{C}}=\mathrm{C}_1 \hat{\mathrm{i}}+\mathrm{C}_2 \hat{\mathrm{j}}+\mathrm{C}_3 \hat{\mathrm{k}}$ $\mathrm{C}_1{ }^2+\mathrm{C}_2{ }^2+\mathrm{C}_3{ }^2=1$ $\overrightarrow{\mathrm{C}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})=|\mathrm{C}| \sqrt{9} \cos 60^{\circ}$ $2 \mathrm{C}_1+2 \mathrm{C}_2-\mathrm{C}_3=\frac{3}{2}$ $\mathrm{C}_1-\mathrm{C}_3=1$ $\mathrm{C}_1+2 \mathrm{C}_2=\frac{1}{2}$ $\mathrm{C}_1=\frac{\sqrt{2}}{3}+\frac{1}{2}$ $\mathrm{C}_2=\frac{-1}{3 \sqrt{2}}$ $\mathrm{C}_3=\frac{\sqrt{2}}{3}-\frac{1}{2}$

Asked in: JEE Main 2024 (04 Apr Shift 1)

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