# Projection of one vector on another?

by Joza
Tags: projection, vector
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,344 Projection of one vector on another? The LENGTH of the projection of one vector onto another is (almost) the dot product. To find the projection of $\vec{u}$ on $\vec{v}$, draw the line from the "tip" of $\vec{u}$ perpendicular with $\vec{v}$. You now have a right triangle with angle $\theta$ between the angles and hypotenuse of length $|\vec{u}|$. The length of the projection, the "near side", is then $|\vec{u}|cos(\theta)$. Since the dot product can be defined as $\vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta)$, to get the length of the pojection, we need to get rid of that $|\vec{v}|$ by dividing by it. The length of the projection of $\vec{u}$ on $\vec{v}$ is $$\frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}$$ In order to get the projection vector itself, we need to multiply that length by the unit vector in the direction of $\vec{v}$, which is, of course, $\vec{v}/|\vec{v}|$. The vector projection of $\vec{u}$ on $\vec{v}$ is $$\frac{\vec{u}\cdot\vec{v}}{|\vec{v}|^2}\vec{v}$$