Let a unit vector which makes an angle of 60° with 2i + 2j - k and an angle of 45° with i - k be C.

Question

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

(1) \(-\frac{\sqrt{2}}{3} \hat{i}+\frac{\sqrt{2}}{3} \hat{j}+\left(\frac{1}{2}+\frac{2 \sqrt{2}}{3}\right) \hat{k}\)

(2) \(\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{1}{2} \hat{\mathrm{k}}\)

(3) \(\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right) \hat{\mathrm{i}}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3 \sqrt{2}}\right) \hat{\mathrm{j}}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right) \hat{\mathrm{k}}\)

(4) \(\frac{\sqrt{2}}{3} \hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{k}}\)

Answer 1

Correct option is (4) \(\frac{\sqrt{2}}{3} \hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{k}}\)

\(\overrightarrow{\mathrm{C}}=\mathrm{C}_{1} \hat{\mathrm{i}}+\mathrm{C}_{2} \hat{\mathrm{j}}+\mathrm{C}_{3} \hat{\mathrm{k}}\)

\(\mathrm{C}_{1}^{2}+\mathrm{C}_{2}^{2}+\mathrm{C}_{3}^{2}=1\)

\(\overrightarrow{\mathrm{C}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})=|\mathrm{C}| \sqrt{9} \cos 60^{\circ}\)

\(2 \mathrm{C}_{1}+2 \mathrm{C}_{2}-\mathrm{C}_{3}=\frac{3}{2}\)

\(\mathrm{C}_{1}-\mathrm{C}_{3}=1\)

\(\mathrm{C}_{1}+2 \mathrm{C}_{2}=\frac{1}{2}\)

\(\mathrm{C}_{1}=\frac{\sqrt{2}}{3}+\frac{1}{2}\)

\(\mathrm{C}_{2}=\frac{-1}{3 \sqrt{2}}\)

\(\mathrm{C}_{3}=\frac{\sqrt{2}}{3}-\frac{1}{2}\)