# What is the projection of (-i + j + k) onto  ( i - j + k)?

Nov 21, 2016

The projection is $\left[\frac{- 1}{3}\right] \left[\hat{i} - \hat{j} + \hat{k}\right]$.
Let us assume $\vec{a} = \left(- \hat{i} + \hat{j} + \hat{k}\right)$ and $\vec{b} = \left(\hat{i} - \hat{j} + \hat{k}\right)$.
The projection of vector '$\vec{b}$' on '$\vec{a}$' $= \left[\frac{\vec{b} . \vec{a}}{| \vec{a} {|}^{2}}\right] \left[\vec{a}\right]$.
$\vec{a} . \vec{b} = {a}_{1} {b}_{1} + {a}_{2} {b}_{2} + {a}_{3} {b}_{3} = - 1$.
$| \vec{a} | = \sqrt{{\left(- 1\right)}^{2} + {\left(1\right)}^{2} + {\left(1\right)}^{2}} = \sqrt{3}$.
$\therefore$ projection of$\vec{b}$ on $\vec{a}$ is $\left[\frac{- 1}{\sqrt{3}} ^ 2\right] \left[\hat{i} - \hat{j} + \hat{k}\right] = \left(- \frac{1}{3}\right) \left(\hat{i} - \hat{j} + \hat{k}\right)$.