$$v_{1} = (1, 1, 1)$$
$$v_{2} = (-1, 1, 0)$$
$$v_{3} = (-1, -1, 2)$$
The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $\mathbb{R}^{3}$.
If $w = (4, -2, 4)$, then what is the coordinate vector, $[w]_{B}$, of $w$ with respect to the basis $B = {v1, v2, v3}$
Suppose
$$(4,-2,4)=w=av_1+bv_2+cv_3\implies \langle v_1,w\rangle = 3a_1$$
Can you now complete the deduction? Can you see why having an orthonormal basis makes things even easier than the above?