3.6 The scalar or dot product
The scalar product, also called dot product
or inner product, of
and
is written as
, and is defined as
|
(3.1) |
This is clearly a number (scalar) and not a
vector. The angle
is the angle between the first and second vector,
and thus
(One usually suppresses the subscript
on the angle
.) We thus see that order does not matter, or more
formally, that the dot product is commutative.
Let us look at some special cases
-
is perpendicular to
. In that case
, and the cosine is zero:
.
-
is parallel to
, i.e.,
.
. For that reason one sometimes writes
for
. Also
-
This is a straighforward application of
the previous two properties! A set where each vector is
orthogonal to all the others is called an
orthogonal set of vectors; if
the vectors also have unit length, one speaks of an
orthonormal set.
It is generally useful to list a few more
properties:
-
. (What is
?)
-
is the product of the
scalar
with the vector
. Thus the result has the same direction as
, with magnitude
.
- We can divide by
since it is a scalar! (Conversely, we can
not divide by a vector!)
-
. (Distributive law). This will not be proven here,
but can easily be done using component form.
Example
3.6:
-
Simplify
Solution:
-
3.6.1 Component form of dot product
Let
,
, then
Example
3.7:
-
Find a unit vector which is
perpendicular to
and has
-component zero.
Solution:
-
This vector has the form
. Must be orthogonal to
, so
which leads to
For this to be a unit vector
, or
(we can choose either sign. Explain!). Thus