Let a, b and c be three non-coplanar unit vectors such that the a… - Vidyadip

MCQ

Let $\vec{a}, \vec{b}$ and $\overrightarrow{ c }$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}=p \vec{a}+q \vec{b}+r \vec{c}$, where $p, q$ and $r$ are scalars, then the value of $\frac{p^2+2 q^2+r^2}{q^2}$ is

A
$1$
B
$2$
C
$3$
✓
$4$

✓

Answer

Correct option: D.

$4$

d
$p \overrightarrow{ a }+ q \vec{b}+ r \overrightarrow{ c }= a \times b + b \times c$

Taking dot product with $\overrightarrow{ a }, \overrightarrow{ b }, \overrightarrow{ c }$ we get

$p+\frac{q}{2}+\frac{r}{2}=[a b c]$ $\quad........(1)$

$\frac{p}{2}+q+\frac{r}{2}=0$ $\quad........(2)$

$\frac{p}{2}+\frac{q}{2}+r=[a b c]$ $\quad........(3)$

$(1)$ and $(3)$ $\Rightarrow p=r$ and $q=-p$

$\frac{p^2+2 q^2+r^2}{q^2}=\frac{p^2+2 p^2+p^2}{p^2}=4 \text { Ans. }$

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