## 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 = [a_{1}, a_{2}] and b = [b_{1},
b_{2}], then

a_{1} = |a| cos θ_{a
}a_{2} = |a| sin θ_{a
}b_{1} = |b| cos θ_{b
}b_{2} = |b| sin θ_{b}

where θ_{a} is the angle of a, and θ_{b}
is the angle of b. Then it follows that

a·b = |a| |b| cos θ

a·b = |a| |b| cos (θ_{a} - θ_{b})

a·b = |a| |b| (cos θ_{a} cos θ_{b}
+ sin θ_{a} sin θ_{b}
)

a·b = |a|cos θ_{a} |b|cos θ_{b}
+ |a|sin θ_{a} |b|sin θ_{b}

a·b = a_{1} b_{1} + a_{2} b_{2}

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

a·b = a_{1} b_{1} + a_{2} b_{2} + a_{3}
b_{3}

### 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.