Consider the following vectors in $\Bbb{R}^3$:

${\bf v}\_1= \frac{1}{\sqrt{3}}(1,1,-1)$, ${\bf v}\_2 = \frac{1}{\sqrt{2}}(1,-1,0),$ and ${\bf v}\_3 = \frac{1}{\sqrt{6}}(1,1,2)$

a) Show that $\{{\bf v}\_1, {\bf v}\_2, {\bf v}\_3\}$ form a basis of $\Bbb{R}^3$. (hint: compute their inner products)

b) Work out the coordinates of a vector ${\bf x} = (x\_1,x\_2,x\_3)$ in the basis $\{{\bf v}\_1, {\bf v}\_2, {\bf v}\_3\}$; that is, find the numbers $c\_1, c\_2, c\_3 \in \Bbb{R}$ such that ${\bf x} = c\_1{\bf v}\_1 + c\_2{\bf v}\_2 +c\_3{\bf v}\_3$.

This is one example in my finals review for linear algebra. I do not know how to tackle this problem especially for part b.