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Q.

The unit vector in ZOX-plane making angles $45^{0}$ and $60^{0}$ respectively with $a = 2 i + 2 j - k$ and $b = j - k$ is

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a

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

b

$\frac{1}{\sqrt{2}} (i - k)$

c

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

d

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

answer is B.

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Detailed Solution

$n = ai + bk |n| = 1 \Rightarrow a^{2} + b^{2} = 1 \frac{\bar{n} . \bar{a}}{|n| |a|} = \cos 45 ° \Rightarrow \frac{2 a - b}{(1) (3)} = \frac{1}{\sqrt{2}} \Rightarrow 2 a - b = \frac{3}{\sqrt{2}} \frac{\bar{n} . \bar{b}}{|n| |b|} = \frac{1}{2} \Rightarrow \frac{- b}{\sqrt{2}} = \frac{1}{2} \Rightarrow b = \frac{1}{\sqrt{2}} 2 a + \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} \Rightarrow 2 a = \frac{2}{\sqrt{2}} \Rightarrow a = \frac{1}{\sqrt{2}} n = \frac{1}{\sqrt{2}} (i - k)$

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