JEE MAIN - Mathematics (2021 - 20th July Morning Shift - No. 16)
Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $$\theta$$, with the vector $$\overrightarrow a $$ + $$\overrightarrow b $$ + $$\overrightarrow c $$. Then 36cos22$$\theta$$ is equal to ___________.
Answer
4
Explanation
$${\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2} = {\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow b } \right|^2} + {\left| {\overrightarrow c } \right|^2} + 2(\overrightarrow a \,.\,\overrightarrow b + \overrightarrow a \,.\,\overrightarrow c + \overrightarrow b \,.\,\overrightarrow c ) = 3$$
$$ \Rightarrow \left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right| = \sqrt 3 \overrightarrow a .(\overrightarrow a + \overrightarrow b + \overrightarrow c ) = \left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|\cos \theta $$
$$ \Rightarrow 1 = \sqrt 3 \cos \theta $$
$$ \Rightarrow \cos 2\theta = - {1 \over 3}$$
$$ \Rightarrow 36{\cos ^2}2\theta = 4$$
$$ \Rightarrow \left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right| = \sqrt 3 \overrightarrow a .(\overrightarrow a + \overrightarrow b + \overrightarrow c ) = \left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|\cos \theta $$
$$ \Rightarrow 1 = \sqrt 3 \cos \theta $$
$$ \Rightarrow \cos 2\theta = - {1 \over 3}$$
$$ \Rightarrow 36{\cos ^2}2\theta = 4$$
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