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 q
where q 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 qa
a2 = |a| sin qa
b1 = |b| cos qb
b2 = |b| sin qb
where qa is the angle of a, and qb
is the angle of b. Then it follows that
a·b = |a| |b| cos q
a·b = |a| |b| cos (qa - qb)
a·b = |a| |b| (cos qa cos qb
+ sin qa sin qb
)
a·b = |a|cos qa |b|cos qb
+ |a|sin qa |b|sin qb
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.
