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Solution for If a Unit Vector Vec A Makes an Angle $\Frac{\Pi}{3} \Text{ with } \Hat{I} , \Frac{\Pi}{4} \Text{ with } \Hat{J}$ and an Acute Angle θ with $\Hat{ K}$ ,Then Find the Value of θ. - CBSE (Commerce) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

Question

If a unit vector  vec a makes an angle $\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}$ and an acute angle θ with $\hat{ k}$ ,then find the value of θ.

Solution

$\text { Since a unit vector makes an angle of } \frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text { with } \hat {j} \text{ and an acute angle } \theta \text{ with } \hat{k} , l = \cos \frac{\pi}{3} \text { or } \frac{1}{2}, m = \cos \frac{\pi}{4}\text { or } \frac{1}{\sqrt{2}} \text { and } n = \cos \theta .$

$\text{ We know }$

$l^2 + m^2 + n^2 = 1$

$\Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{\sqrt{2}} \right)^2 + \cos^2 \theta = 1$

$\Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \theta = 1$

$\Rightarrow \cos^2 \theta = \frac{1}{4}$

$\Rightarrow \cos \theta = \frac{1}{2}$

$\Rightarrow \theta = \frac{\pi}{3}$

$\text { Thus, the vector } \vec{a} \text { makes an angle of } \frac{\pi}{3} \text { with } \hat {k} .$

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Solution If a Unit Vector Vec A Makes an Angle $\Frac{\Pi}{3} \Text{ with } \Hat{I} , \Frac{\Pi}{4} \Text{ with } \Hat{J}$ and an Acute Angle θ with $\Hat{ K}$ ,Then Find the Value of θ. Concept: Direction Cosines and Direction Ratios of a Line.
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