How do I find the orthogonal projection of a vector?

Oct 19, 2014

The orthogonal projection of $\vec{a}$ onto $\vec{b}$ can be found by

$\left(\vec{a} \cdot \frac{\vec{b}}{|} \vec{b} |\right) \frac{\vec{b}}{|} \vec{b} | = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}$

Let us find the orthogonal projection of $\vec{a} = \left(1 , 0 , - 2\right)$ onto $\vec{b} = \left(1 , 2 , 3\right)$.

$\frac{\left(1 , 0 , - 2\right) \cdot \left(1 , 2 , 3\right)}{\left(1 , 2 , 3\right) \cdot \left(1 , 2 , 3\right)} \left(1 , 2 , 3\right) = \frac{- 5}{14} \left(1 , 2 , 3\right) = \left(- \frac{5}{14} , - \frac{10}{14} , - \frac{15}{14}\right)$.

I hope that this was helpful.