If $\bar{a} = 3 \hat{i} - \hat{j} + 5 \hat{k}$ and $\bar{b} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ are given vectors. A vector $\bar{c}$ which is perpendicular to Z-axis satisfying $\bar{c} \cdot \bar{a} = 9$ and $\bar{c} \cdot \bar{b} = - 4$ . If inclination of with X-axis and Y-axis is $α$ and $β$ respectively, then which of the following is not true ?

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$\cos^{2} α + \cos^{2} β = 1$

$α > \frac{π}{2}$

$α > \frac{π}{4}$

$β > \frac{π}{2}$

answer is A.

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Detailed Solution

$c$ lies in XY-plane
$\Rightarrow c = x \hat{i} + y \hat{j}$
from the given condition 3 x - y = 9 and x + 2y = - 4
solving we get $c = 2 \hat{i} - 3 \hat{j}$
$α = \cos^{- 1} (\frac{2}{\sqrt{13}}), β = \cos^{- 1} (\frac{- 3}{\sqrt{13}})$

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