Length of projection, Projection vector, Perpendicular distance from a point to vector

Length of projection & Projection vector

The length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$ is given by $$ \boxed{ \left| \overrightarrow{ON} \right| = \left| \textbf{a} \cdot \hat{ \textbf{b}} \right| } $$

The projection vector of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$ is given by $$ \boxed{ \overrightarrow{ON} = \left( \textbf{a} \cdot \hat{ \textbf{b}} \right) \hat{ \textbf{b} } } $$

If the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is an obtuse angle, the following diagram is more appropriate (formulas remain unchanged):

Perpendicular distance from point to vector

The perpendicular distance from point $A$ to $\overrightarrow{OB}$ is given by $$ \boxed{ \left| \overrightarrow{AN} \right| = \left| \textbf{a} \times \hat{ \textbf{b}} \right| } $$

The perpendicular distance is also the shortest distance from point $A$ to $\overrightarrow{OB}$. Besides using the formula above, we can use Pythagoras theorem to find the distance: $$ AN = \sqrt{ OA^2 - ON^2 } $$

Recall that $ON$ is the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.