Let a, b and c be three unit vectors such that
Question:

Let $a, b$ and $c$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ is equal to__________.

Solution:

$|\vec{a}|=|\vec{b}|=|\vec{c}|=1$

$|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$

$\Rightarrow \vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}=-2$

Now, $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$

$=2|\vec{a}|^{2}+4|\vec{b}|^{2}+4|\vec{c}|^{2}+4(\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c})=2$