A unit vector in $x y$ -plane that makes an angle of $45^{0}$ with the vector $\bar{i} + \bar{j}$ and an angle of $60^{0}$ with the vector $3 \bar{i} - 4 \bar{j}$ is

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$\bar{i}$

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

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

Cannot be determined

answer is D.

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Detailed Solution

Let the unit vector in the xy-plane be
$\bar{c} = x \hat{i} + y \hat{j} \Rightarrow | \bar{c} | = 1 \Rightarrow x^{2} + y^{2} = 1 - - - - - - - - (1)$
Given $\cos 45^{0} = \frac{1}{\sqrt{2}} = \frac{(x \hat{i} + y \hat{j}) (\hat{i} + \hat{j})}{\sqrt{x^{2} + y^{2}} \sqrt{1 + 1}} \Rightarrow x + y - - - - - - - - (2)$
$\cos 60^{0} = \frac{1}{2} = \frac{(x \hat{i} + y \hat{j}) (3 \hat{i} - 4 \hat{j})}{\sqrt{x^{2} + y^{2}} \sqrt{9 + 16}} \Rightarrow 3 x + 5 y = \frac{5}{2} - - - - - - - - (3)$
Solving (2)&(3) $x = \frac{13}{14}; y = \frac{1}{14}$ these values do not satisfy (1)