vector algebra results/formulae

just some stuff to memorise for ease.

commutative

order of operation doesn’t matter

associative

order of brackets does not matter

properties of scalar dot product

commutative

distributive for addition: a . (b + c) = a.b + a.c

a.b = 0 for perpendicular vectors

projection of vector b onto vector a

the ‘shadow’ of b on a

properties of vector cross product

scalar triple product

this is also the volume of a parallelpiped whose edges are a, b, c.

= 0 if a, b, c are coplanar

vector triple product

equations in space

shortest distance from line to origin (scalar)

t is unit vector in direction of line, and a is a known point on the line

shortest distance from line to origin (vector)

shortest distance from point P to a line (scalar)

p is vector from origin to point P

shortest distance from line to point P (vector)

shortest distance from a line to a line

a and b are vectors to known points on lines 1 and 2

n_hat is the unit vector perpendicular to both a and b:

n = s x t / |s x t|

shortest distance from point to a plane

a is known point on the plane

n_hat is vector perpendicular to plane

this equation can also be applied to line and plain, with p being known point on line

reciprocal vectors

when a set of vectors and their reciprocals are perpendicular to the other vectors they aren’t matched with.

projection of vector area A onto a plane

n_hat is the normal vector to the plane