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Dot Product

The dot product is also called the "scalar product" because its a scalar (a single value; not a vector).

The dot product of a and b, a·b is defined as follows:

a·b = |a| |b| cos θ

where θ is the angle between a and b, when placed so that their tails coincide.

In two dimensions, if a = [a1, a2] and b = [b1, b2], then

a1 = |a| cos θa
a2 = |a| sin θa
b1 = |b| cos θb
b2 = |b| sin θb

where θa is the angle of a, and θb is the angle of b.  Then it follows that

a·b = |a| |b| cos θ  
a·b = |a| |b| cos (θa - θb)  
a·b = |a| |b| (cos θa cos θb + sin θa sin θb )  
a·b = |a|cos θa |b|cos θb + |a|sin θa |b|sin θb  
a·b = a1 b1 + a2 b2  

This result generalizes to any number of dimensions, so in 3 dimensions, for example,

a·b = a1 b1 + a2 b2 + a3 b3 

Properties of the Dot Product

Commutative: yes

a·b = b·a

Associative: no

(a·b)·c doesn't even make sense, because the two operands of the dot product must be vectors, but the result is a scalar.  However, the "triple product" is a combination of the cross product and dot product that represents the volume of an oblique rectangular solid.

Triple Product

(a�b)·c is a scalar representing the "signed volume" of a parallelepiped (a 6-faced polyhedron all of whose faces are parallelograms lying in pairs of parallel planes) whose dimensions and angles are given by the three vectors a, b, and c.  If the vectors represent three edges of the solid that meet at a point, and they are named in counterclockwise order from the point of view of the interior of the solid (i j k order, if you will), then the volume will be positive.  Otherwise negative.

The triple product is, in its way, commutative.  That is, any ordering of a, b, and c give a result with the same absolute value, with its sign determined by the order the edges are named.

Related Pages in this website

Triangle Area Using Vectors, part 1

Vector Cross Product

Triple Product -- (a�b)·c is a scalar representing the "signed volume" of a parallelepiped

Triangle Area using Determinant

Matrix Math

Geometry and Trigonometry, and in particular, the Points and Lines section.

The webmaster and author of this Math Help site is Graeme McRae.