Vector projection

Parametric equations

Any vector $\mathbf{a}$ can be split, relative to a nonzero vector $\mathbf{b}$, into a parallel part ${\mathbf{a}}_{\parallel }$ and a perpendicular part ${\mathbf{a}}_{\perp }$, with $\mathbf{a}={\mathbf{a}}_{\parallel }+{\mathbf{a}}_{\perp }$ and ${\mathbf{a}}_{\perp }\cdot \mathbf{b}=0$. The projection is what the dot product measures geometrically. It is also the least-squares best approximation of $\mathbf{a}$ along $\mathbf{b}$: $|{\mathbf{a}}_{\perp }|$ is the shortest distance from the tip of $\mathbf{a}$ to the line through $\mathbf{b}$.

If $\mathbf{b}=0$, the projection is undefined. The same split works with the roles reversed, projecting $\mathbf{b}$ onto $\mathbf{a}$.

In the basis reading, ${\mathbf{e}}_1$ is the unit vector along the line and ${\mathbf{e}}_2$ is ${\mathbf{e}}_1$ turned $90^{\circ }$ counterclockwise. They are orthonormal, and the same vector is $c_1{\mathbf{e}}_1+c_2{\mathbf{e}}_2$.

Interaction

• a onto b: show the foot of the perpendicular from $\mathbf{a}$ to the line of $\mathbf{b}$, and the two parts. b onto a reverses the roles
• Drop animation: a point moves between the tip and the foot, so the parallel and perpendicular parts separate
• Error length: mark $|{\mathbf{a}}_{\perp }|$, the shortest distance from the tip to the line
• Decomposition: the parallel and perpendicular parts. Basis e1, e2 reads the same vector in that orthonormal basis

Related

• Geometric significance of the dot product
• Vector addition and scalar multiplication
• Basic patterns of a vector field

Further reading

• Vector projection (Wikipedia)