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

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$
$\lambda=\frac{1}{2}$
$\therefore 2\left|\left(\frac{-3}{2}-1+2\right) \hat{ i }+\left(\frac{5}{2}+1\right) \hat{ j }+(3+1+1) \hat{ k }\right|^{2}$
$=2\left(\frac{1}{4}+\frac{49}{4}+25\right)=25+50=75$