Question

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}\cdot \vec{b}=0$. For some $x,y\in$ ℝ let $\vec{c}=x\vec{a}+y\vec{b}+\left(\vec{a}\times \vec{b}\right)$. If $\left|\vec{c}\right|=2$
and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then the value of $8co{s}^2\alpha$ is _____ .

Answer 1

Answer: 3

$\vec{c}=x\vec{a}+y\vec{b}+\vec{a}\times \vec{b}$
$\vec{c}\cdot \vec{a}=x$ and $x=2\cos \alpha$
$\vec{c}\cdot \vec{a}=y$ and $y=2\cos \alpha$
Also
$|\vec{a}\times \vec{b}|=1$
$\therefore \vec{c}=2\cos (\vec{a}+\vec{b})+\vec{a}\times \vec{b}$
${\vec{c}}^2=4{\cos }^2\alpha (\vec{a}+\vec{b}{)}^2+(\vec{a}+\vec{b}{)}^2$
$+2\cos \alpha (\vec{a}+\vec{b}).(\vec{a}\times \vec{b})$
$4=8{\cos }^2\alpha +1$
$8{\cos }^2\alpha =3$