The scalar product

Definition

The dot product of two vectors ->u and ->v, denoted ->u . ->v, is a real number (a scalar, not a vector). It can be calculated in several ways.

Using coordinates

If ->u(x; y) and ->v(x'; y'), then:

->u . ->v = xx' + yy'

Example: ->u(2; 3) and ->v(4; -1) give ->u·->v = 2·4 + 3·(-1) = 8 - 3 = 5.

Using the magnitude and angle

->u· ->v = ||u|| * ||v|| * cos(angle(u,v))

This formula is useful in trigonometry, particularly for calculating angles in a triangle.

Orthogonal vectors

Two vectors are orthogonal if and only if their dot product is zero:

->u · ->v = 0 <=> ->u is perpendicular to ->v

This is the quickest way to prove that two lines are perpendicular.

Useful properties

• ->u · ->v = ->v · ->u (symmetry)
• ->u . ->u = ||u||² (scalar square)
• ->u . (->v + ->w) = ->u . ->v + ->u . ->w (distributivity)

Common pitfall

The dot product yields a NUMBER, never a vector: do not write “->u·->v = ->w”. Furthermore, ->u·->v = 0 does not imply that ->u or ->v is the zero vector: it merely means that they are perpendicular.