Let a=i+ j+ k, , R. Let a vector b be such that the angle between… - Vidyadip

MCQ

Let $\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in R$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2=6$, If $\vec{a} \cdot \vec{b}=3 \sqrt{2}$, then the value of $\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2$ is equal to

A
90
B
75
C
95
D
85

✓

Answer

$|\overrightarrow{b}|^2=6 ;|\overrightarrow{a}||\overrightarrow{b}| \cos \theta=3 \sqrt{2}$
$|\overrightarrow{a}|^2|\overrightarrow{b}|^2 \cos ^2 \theta=18$
$|\overrightarrow{a}|^2=6$
Also $1+\alpha^2+\beta^2=6$
$\alpha^2+\beta^2=5$
to find $\left(\alpha^2+\beta^2\right)|\vec{a}|^2|\vec{b}|^2 \sin ^2 \theta$
$=(5)(6)(6)\left(\frac{1}{2}\right)$
$=90$

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