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2.2. The dot product of two vectors 23
2.2 The dot product of two vectors
The dot product is used to project a vector in a given direction, to reduce a vector
to components, to reduce vector equations to scalar equations, to define work and
power, and to help solve geometry problems.
The dot product of two vectors
A and
B is written
A ·
B (pronounced ‘A dot B’).
The dot product of
A and
B is the product of the magnitudes of the two vectors times
a number that expresses the degree to which
A and
B are parallel: cos θAB, where
θAB is the angle between
A and
B. That is,
B
A
θAB
B
A
cosθAB
cosθAB
Figure 2.17: The dot product of
A and
B is a scalar and so is not easily drawn. It is
given by
A ·
B = AB cos θAB which is A
times the projection of
B in the A direction
and also B times the projection of
A in the
B direction.
(Filename:tfigure1.11)
A ·
B def = |
A| |
B| cos θAB
which is sometimes written more concisely as
A ·
B = AB cos θ. One special case
is when cos θAB = 1,
A and
B are parallel, and
A ·
B = AB. Another is when
cos θAB = 0,
A and
B are perpendicular, and
A ·
B = 0.
1
1 If you don’t know, almost without
a thought, that cos 0 = 1, cos π/2 =
0,sin 0 = 0, and sin π/2 = 1 now is as
good a time as any to draw as many triangles and unit circles as it takes to cement
these special cases into your head.
The dot product of two vectors is a scalar. So the dot product is sometimes called
the scalar product. Using the geometric definition of dot product, and the rules for
vector addition we have already discussed, you can convince yourself of (or believe)
the following properties of dot products.
•
A ·
B =
B ·
A commutative law,
AB cos θ = B A cos θ
• (a
A) ·
B =
A · (a
B) = a(
A ·
B) a distributive law,
(a A)B cos θ = A(aB) cos θ
•
A · (
B +
C) =
A ·
B +
A ·
C another distributive law,
the projection of
B +
C onto
A is the
sum of the two separate projections
•
A ·
B = 0 if
A ⊥
B perpendicular vectors have zero for
a dot product, AB cos π/2 = 0
•
A ·
B = |
A||
B| if
A
B parallel vectors have the product of
their magnitudes for a dot product,
AB cos 0 = AB. In particular,
A ·
A = A2 or |
A| = √
A ·
A
• ıˆ · ıˆ = ˆ · ˆ = kˆ · kˆ = 1,
ıˆ · ˆ = ˆ · kˆ = kˆ · ıˆ = 0
The standard base vectors used with
cartesian coordinates are unit vectors
and they are perpendicular to each
other. In math language they are ‘orthonormal.’
• ıˆ
· ıˆ
= ˆ
· ˆ
= kˆ
· kˆ
= 1,
ıˆ
· ˆ
= ˆ
· kˆ
= kˆ
· ıˆ
= 0
The standard crooked base vectors
are orthonormal.
The identities above lead to the following equivalent ways of expressing the dot
product of
A and
B (see box 2.2 on page 24 to see how the component formula
follows from the geometric definition above):