3.9 The vector product
We have now looked extensively at the
scalar product, and now look at the vector product, that
returns a vector. Two standard notations are used
|
(3.2) |
We shall use the first notation. Other
terms used are “cross product” or “outer
product”.
The vector product of two vectors
and
is defined as a vector, see Fig.
3.15
,
- of magnitude
- of a direction orthogonal to both
and
, so that
,
and
form a right-handed set
The magnitude of the outer product is exactly
equal to the area of the parallelogram with sides
and
,
. calculation of the outer product in component form (to
be discussed below) is thus an easy way to obtain this
area.
Let
be a unit vector in the direction of
, then
. From the right handed rule we see that
, i.e., the vector product is
not commutative. Properties of the
outer product:
- For parallel vectors
and so
, in particular
.
- For orthogonal
vectord, i.e., the angle
between
and
is
, any two of the vectors
,
and
are orthogonal.
- The coordinate vectors
,
,
:
- From
we see that
.
-
. Follows most easily from component form (see
below).
- Component form:
Using
and similar for
, we find
This last line is often summarized in the
form of a determinant
Example
3.10:
-
Give
and
, find
.
Solution:
-
Example
3.11:
-
Find
given
, and
,and find
the unit vector perpendicular to
and
.
Expand by Row 1: and we get
.
Other examples: