Let a unit vector which makes an angle of 60° with and an angle of 45° with be . Then is [2024]

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Solution

Q.
Let a unit vector which makes an angle of 60° with $2 \hat{i} + 2 \hat{j} - \hat{k}$ and an angle of 45° with $\hat{i} - \hat{k}$ be $\vec{C}$ . Then $\vec{C} + (- \frac{1}{2} \hat{i} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k})$ is [2024]

1 $\frac{\sqrt{2}}{3} \hat{i} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{1}{2} \hat{k}$

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

3 $\frac{\sqrt{2}}{3} \hat{i} - \frac{1}{2} \hat{k}$

4 $(\frac{1}{\sqrt{3}} + \frac{1}{2}) \hat{i} + (\frac{1}{\sqrt{3}} - \frac{1}{3 \sqrt{2}}) \hat{j} + (\frac{1}{\sqrt{3}} + \frac{\sqrt{2}}{3}) \hat{k}$

Ans.
(3)

Let $\vec{C} = a \hat{i} + b \hat{j} + c \hat{k}$ , then we have $| \vec{C} |$ $= 1$

$\Rightarrow a^{2} + b^{2} + c^{2} = 1$ ... (i)

Angle between $\vec{C}$ and $2 \hat{i} + 2 \hat{j} - \hat{k}$ is 60°.

$∴ (a \hat{i} + b \hat{j} + c \hat{k}) \cdot (2 \hat{i} + 2 \hat{j} - \hat{k}) = (\sqrt{4 + 4 + 1}) \cos 60 °$ $(∵ a^{2} + b^{2} + c^{2} = 1)$

$\Rightarrow 2 a + 2 b - c = \frac{3}{2}$ ... (ii)

Angle between $\vec{C}$ and $\hat{i} - \hat{k}$ is 45°.

$∴ (a \hat{i} + b \hat{j} + c \hat{k}) \cdot (\hat{i} - \hat{k}) = \sqrt{1 + 0 + 1} \cos 45 °$

$\Rightarrow a - c = 1$ ... (iii)

Solving (i), (ii) and (iii), we get $a + 2 b = \frac{1}{2}$ and $a^{2} + b^{2} + {(a - 1)}^{2} = 1$

$\Rightarrow 2 a^{2} - 2 a + b^{2} = 0 \Rightarrow 2 a^{2} - 2 a + {(\frac{2 a - 1}{4})}^{2} = 0$

$\Rightarrow 32 a^{2} - 32 a + 4 a^{2} - 4 a + 1 = 0 \Rightarrow 36 a^{2} - 36 a + 1 = 0$

$\Rightarrow a = \frac{36 \pm \sqrt{{(36)}^{2} - 4 (36)}}{2 \times 36} = \frac{1}{2} \pm \frac{\sqrt{2}}{3}$

$∴ b = \frac{1 - 2 a}{4} \Rightarrow b = \frac{1 - 1 \mp \frac{2 \sqrt{2}}{3}}{4} = \mp \frac{1}{3 \sqrt{2}}$

$∴ c = - \frac{1}{2} \pm \frac{\sqrt{2}}{3}$

$∴ \vec{C} + (\frac{- 1}{2} \hat{i} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k}) = \frac{\sqrt{2}}{3} \hat{i} - \frac{1}{2} \hat{k}$

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