JEE MAIN - Mathematics (2019 - 9th April Evening Slot - No. 7)
If a unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$ , $$\pi $$/ 4
with $$\widehat j$$ and $$\theta $$$$ \in $$(0, $$\pi $$) with $$\widehat k$$, then a value of $$\theta $$
is :-
$${{5\pi } \over {6}}$$
$${{5\pi } \over {12}}$$
$${{2\pi } \over {3}}$$
$${{\pi } \over {4}}$$
Explanation
A unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$
$$ \therefore $$ $$\alpha $$ = $$\pi $$/3
and $$\pi $$/ 4 with $$\widehat j$$
$$ \therefore $$ $$\beta $$ = $$\pi $$/ 4
and $$\theta $$$$ \in $$(0, $$\pi $$) with $$\widehat k$$
$$ \therefore $$ $$\gamma $$ = $$\theta $$
We also know,
$${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma $$ = 1
$$ \Rightarrow $$ $${\cos ^2}{\pi \over 3} + {\cos ^2}{\pi \over 4} + {\cos ^2}\theta $$ = 1
$$ \Rightarrow $$ $${1 \over 4} + {1 \over 2} + {\cos ^2}\theta $$ = 1
$$ \Rightarrow $$ $${\cos ^2}\theta $$ = $$ \pm {1 \over 2}$$
$$ \Rightarrow $$ $$\theta $$ = $${\pi \over 3}$$ or $${{2\pi } \over {3}}$$
$$ \therefore $$ $$\alpha $$ = $$\pi $$/3
and $$\pi $$/ 4 with $$\widehat j$$
$$ \therefore $$ $$\beta $$ = $$\pi $$/ 4
and $$\theta $$$$ \in $$(0, $$\pi $$) with $$\widehat k$$
$$ \therefore $$ $$\gamma $$ = $$\theta $$
We also know,
$${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma $$ = 1
$$ \Rightarrow $$ $${\cos ^2}{\pi \over 3} + {\cos ^2}{\pi \over 4} + {\cos ^2}\theta $$ = 1
$$ \Rightarrow $$ $${1 \over 4} + {1 \over 2} + {\cos ^2}\theta $$ = 1
$$ \Rightarrow $$ $${\cos ^2}\theta $$ = $$ \pm {1 \over 2}$$
$$ \Rightarrow $$ $$\theta $$ = $${\pi \over 3}$$ or $${{2\pi } \over {3}}$$
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