10
A
REVIEW OF
VECTOR
AND
MATRIX
ALGEBRA
over
j
sums
over the columns. Thus, Equation
1.38
in matrix notation is
100
001
x.B-+[A]t[l][B]
=
[A1
A2
A31
[0
1
01
[ii]
=
[AIt[B].
(1.46)
The
Cross
Product
The cross or vector product between two vectors
a third vector
c,
which
is
written andB forms
-
C=AXB.
(
1.47)
The magnitude of the vector
e
is
where
8
is
the angle between the two vectors,
as
shown in Figure
1.4.
The direction
of
c
depends on the “handedness” of the coordinate system. By convention, the
three-dimensional coordinate system
in
physics
are
usually “right-handed.’’ Extend
the fingers of your right hand
straight,
aligned along the basis vector
61.
Now, curl
your fingers toward
&he
basis
vector
$2.
If
your
thumb
now points along
63,
the
coordinate system
is
right-handed. When the coordinate system
is
arranged
this
way,
the direction
of
the
cross
product follows a
similar
rule.
To
determine the direction
of
C
in Equation
1.47,
point your fingers along
A,
and curl them
to
point along
B.
Your
thumb
is
now pointing
in
the direction
of
e.
This
definition
is
usually called the
right-
hand
mle.
Notice, the direction of is always perpendicular to the plane formed
by
A
and
B.
If,
for some reason, we
are
using a left-handed coordinate system,
the definition for the cross product changes, and we must instead use a
left-hand
rule.
Because the definition of a cross product changes slightly when we move to
-
I
I
,
\
\
\
\
,
\
\
‘.
‘
Figure
1.4
The
Cross
Product
-1
