Question

Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b},\vec{a}\cdot \vec{c}=7$ and $\vec{b}$ is perpendicular to $\vec{c},$ where $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{k},$ then the value of $2|\vec{a}+\vec{b}+\vec{c}{|}^2$ is _______.

Answer 1

Answer: 75

Let $\vec{c}=\lambda (\vec{b}\times (\vec{a}\times \vec{b}))$
$=\lambda ((\vec{b}\cdot \vec{b})\vec{a}-(\vec{b}\cdot \vec{a})\vec{b})$
$=\lambda (5(-\hat{i}+\hat{j}+\hat{k})+2\hat{i}+\hat{k})$
$=\lambda (-3\hat{i}+5\hat{j}+6\hat{k})$
$\vec{c}\cdot \vec{a}=7$
$\Rightarrow 3\lambda +5\lambda +6\lambda =7$
$\lambda =\frac{1}{2}$
$\therefore 2{\left|\left(\frac{-3}{2}-1+2\right)\hat{i}+\left(\frac{5}{2}+1\right)\hat{j}+(3+1+1)\hat{k}\right|}^2$
$=2\left(\frac{1}{4}+\frac{49}{4}+25\right)=25+50=75$