map ((+) 1) (2,3) == (3,4)

map2 (*) (2,4) (3,2) == (6,8)

foldl (+) 0 (2,4) == 6

foldr (-) 0 (1,12) == -11

v + w

add (1,2) (4,5) == (5,7)

v - w

sub (3,1) (-3,8) == (6,-7)

-v

negate (2,-4) == (-2,4)

a*v

scale 3 (2,3) = (6,9)

v/a

divideBy 4 (12,16) == (3,4)

v dot w

The dot product of two vectors. Also called scalar product or inner product.

It links the length and angle of two vectors.

v dot w = |v|*|w|*cos(phi)

dot (1,2) (3,2) == 1*3 + 2*2 == 7

The length of a vector. Also known as magnitude or norm.

|v| = sqrt(v dot v)

length (3,4) == sqrt(3^2+4^2) == 5

The squared length. This is cheaper to calculate, so if you only need to compare lengths you can use this instead of the length.

|v|^2 = v dot w

lengthSquared (3,4) == 3^2+4^2 == 25

Normalizes a vector. This will give you a unit vector (e.g. with length 1) in the same direction as v.

v/|v|

normalize (3,4) == (3/5,4/5)

A unit vector pointing from v to w

(w - v)/|w - v|

directionFromTo (5,1) (8,5) == (3/5,4/5)

Calculates the distance from v to w.

|v - w| = |w - v|

distance (3,0) (0,4) == 5

The squared distance. This is slightly faster.

|v - w|^2

distanceSquared (3,0) (0,4) == 25

The angle between two vectors. The angle is in radians.

acos((v dot w)/(|v|*|w|))

angle (1,0) (2,2) == pi/4    -- or 45°

The projection of v onto w.

(v dot w)/|w| * w/|w|

project (2,1) (4,0) == (2,0)

The rejection of v onto w. This is always perpendicular to the projection.

v - (project v w)

reject (2,1) (4,0) == (0,1)