Let a unit vector make angles and with the vectors , and respectively. If , then is equal to [2024]

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Solution

Q.
Let a unit vector $\hat{u} = x \hat{i} + y \hat{j} + z \hat{k}$ make angles $\frac{π}{2}, \frac{π}{3}$ and $\frac{2 π}{3}$ with the vectors $\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{k}$ , $\frac{1}{\sqrt{2}} \hat{j} + \frac{1}{\sqrt{2}} \hat{k}$ and $\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}$ respectively. If $\vec{v} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} + \frac{1}{\sqrt{2}} \hat{k}$ , then $| \hat{u} - \vec{v} |^{2}$ is equal to [2024]

1 9

2 $\frac{11}{2}$

3 7

4 $\frac{5}{2}$

Ans.
(4)

Let $\vec{a} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{k}$ , $\vec{b} = \frac{1}{\sqrt{2}} \hat{j} + \frac{1}{\sqrt{2}} \hat{k}$ and $\vec{c} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}$

Now, $\hat{u} \cdot \vec{a} = | \hat{u} | \cdot | \vec{a} | \cos θ$ ,

$\Rightarrow \frac{x}{\sqrt{2}} + \frac{z}{\sqrt{2}} = 1 \cdot 1 \cdot \cos \frac{π}{2} \Rightarrow x + z = 0$ ... (i)

Again $\hat{u} \cdot \vec{b} = \frac{y}{\sqrt{2}} + \frac{z}{\sqrt{2}} = 1 \cdot 1 \cdot \cos \frac{π}{3} \Rightarrow y + z = \frac{1}{\sqrt{2}}$ ... (ii)

and $\hat{u} \cdot \vec{c} = \frac{x}{\sqrt{2}} + \frac{y}{\sqrt{2}} = 1 \cdot 1 \cdot \cos \frac{2 π}{3} \Rightarrow x + y = \frac{- 1}{\sqrt{2}}$ ... (iii)

Solving (i), (ii) and (iii), we get

$x = \frac{- 1}{\sqrt{2}}, y = 0, z = \frac{1}{\sqrt{2}} ∴ \hat{u} = \frac{- 1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{k}$

So, $| \hat{u} - \vec{v} |^{2} =$ $| - \sqrt{2} \hat{i} - \frac{1}{\sqrt{2}} \hat{j} |^{2}$ $= 2 + \frac{1}{2} = \frac{5}{2}$ .

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