Let a¯=xi¯+yj¯+zk¯ and x=2y. |a¯|=52 and a¯ makes an angle of 135° with z-axis then a¯ =

cos⁡γ=z|a¯|⇒z=52×-12=-5|a¯|=52⇒x2+y2+z2=50⇒5y2=25⇒y=5a¯=25i¯+5j¯−5k¯

Let a¯=xi¯+yj¯+zk¯ and x=2y. |a¯|=52 and a¯ makes an angle of 135° with z-axis then a¯ =

cos⁡γ=z|a¯|⇒z=52×-12=-5|a¯|=52⇒x2+y2+z2=50⇒5y2=25⇒y=5a¯=25i¯+5j¯−5k¯

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Let $\bar{a} = x \bar{i} + y \bar{j} + z \bar{k}$ and x=2y. $| \bar{a} | = 5 \sqrt{2}$ and $\bar{a}$ makes an angle of 135° with z-axis then $\bar{a}$ =

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$2 \sqrt{3} \bar{i} + \sqrt{3} \bar{j} - 3 \bar{k}$

$2 \sqrt{6} \bar{i} + \sqrt{6} \bar{j} - 6 \bar{k}$

$2 \sqrt{5} \bar{i} + \sqrt{5} \bar{j} - 5 \bar{k}$

$2 \sqrt{5} \bar{i} + \sqrt{5} \bar{j} + 5 \bar{k}$

answer is C.

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Detailed Solution

$\begin{array}{l} \cos γ = \frac{z}{| \bar{a} |} \Rightarrow z = 5 \sqrt{2} \times \frac{- 1}{\sqrt{2}} = - 5 \\ | \bar{a} | = 5 \sqrt{2} \Rightarrow x^{2} + y^{2} + z^{2} = 50 \\ \Rightarrow 5 y^{2} = 25 \Rightarrow y = \sqrt{5} \\ \bar{a} = 2 \sqrt{5} \bar{i} + \sqrt{5} \bar{j} - 5 \bar{k} \end{array}$