Parallel and perpendicular component

Every vector $\vec a$ can be decomposed into the sum of a part parallel to a given direction $\vec b$ and a part perpendicular to it. It is the mechanism underlying the computation of the point–line distance.

Property — Parallel and perpendicular component

$\vec a_\parallel = \frac{\vec a\cdot\vec b}{|\vec b|^2}\,\vec b, \qquad \vec a_\perp = \vec a - \vec a_\parallel.$ The distance of $\vec a$ from the direction of $\vec b$ is $|\vec a_\perp|$.

Example

Let us decompose $\vec a = (-7;0;2)$ with respect to the unit vector $\hat v\approx(0{,}371;\, 0{,}557;\, -0{,}742)$ (unit vector of $(2;3;-4)$).

Scalar projection: $\vec a\cdot\hat v = -7\cdot 0{,}371 + 0 + 2\cdot(-0{,}742) = -2{,}597 - 1{,}484 \approx -4{,}08.$ Hence $\vec a_\parallel = -4{,}08\cdot\hat v$ and $\vec a_\perp = \vec a - \vec a_\parallel$; the length $d = |\vec a_\perp|$ is the distance from the point to the line with direction $\vec v$.

Using the unit vector $\hat v$ directly (of modulus $1$) the formula simplifies to $\vec a_\parallel = (\vec a\cdot\hat v)\,\hat v$.

Links

Topics: Analytic geometry in space
Concepts: Parallel component · Scalar product · Vector
Skills: Analytic geometry · Using formulae