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 $8\, cos^{2} \alpha$ is _____ .

Answer 1

$\vec{c}=x \vec{a}+y \vec{b}+\vec{a} \times \vec{b}$
$\vec{c} \cdot \vec{a}=x$ 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 $
$8 \cos ^{2} \alpha=3$