If |a|=3 ;|b|=5 ; b c=10, angle between b and c is 3 , a is perpe… - Vidyadip

MCQ

If $|\vec{a}|=3 ;|\vec{b}|=5 ; \vec{b} \cdot \vec{c}=10$, angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}, \vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$. Then the value of $|\vec{a} \times(\vec{b} \times \vec{c})|$ is

A
20
B
30
✓
60
D
40

✓

Answer

Correct option: C.

60

(c) : Given $\vec{b} \cdot \vec{c}=10$ and angle between
$\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$
$
\Rightarrow|\vec{b}||\vec{c}| \cos \frac{\pi}{3}=10 \Rightarrow 5|\vec{c}| \cdot \frac{1}{2}=10 \Rightarrow|\vec{c}|=4
$
Now, $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$.
$\Rightarrow$ Angle between $\vec{a}$ and $\vec{b} \times \vec{c}$ is $\frac{\pi}{2}$.
Now, consider $|\vec{a} \times(\vec{b} \times \vec{c})|=|\vec{a}||\vec{b} \times \vec{c}| \sin \frac{\pi}{2}$
$
=3 \times|\vec{b}||\vec{c}| \sin \frac{\pi}{2} \times 1=3 \times 5 \times 4 \times 1=60
$

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