SECTION 1.5
THE CROSS PRODUCT
17
1.5.
T
HE
C
ROSS
P
RODUCT
As we saw in the last section, it is easiest to write down an equation for a plane if we
know its normal vector. In order to Fnd the equation of a plane containing two given
non-skew lines, then, we want to be able to Fnd a vector orthogonal to both lines. In this
section we introduce a new operation on vectors, called the
cross product
, and written
a
b
,
which does this (and more). Instead of the usual approach of presenting the somewhat
mysterious deFnition of the cross product and then studying its properties, we are going
to go in the other direction. We are going to write down our goals for this operation, and
then use these to Fgure out the formula. Since all of these goals are going to turn out to
hold for the cross product, we are going to call them properties, but the reader should keep
in mind that they are just desires until we prove them.
Unlike the dot product which produces a scalar, the cross product will produce a
vector
.
We will create the cross product so that this new vector,
a
b
, is orthogonal to both
a
and
b
, but this raises a question: which direction? ±or example, should
i
j
be
k
or
k
? Let’s
decide that our goal will be to have the cross product agree with the right-hand rule (or
book rule). This means that if you point you right arm in the direction of
a
and then curl
your Fngers in towards your palm in the direction of
b
(at an angle less than
180
), then
your thumb points in the direction of
a
b
.
Orthogonality Property of Cross Products.
The cross product
a
b
is orthogonal to both
a
and
b
, and points in the direction given by
the right-hand rule.
Because we are insisting on this property, note that the cross product will
not
be com-
mutative. Going back to
i
and
j
:
i
j
k
, but
j
i
k
. In fact, the cross product will
turn out to always be
anti-commutative
:
Anti-Commutativity Property of Cross Products.
±or all vectors
a
and
b
,
a
b
b
a
.
It is good practice to try all the cross products of basis vectors at this point. One will
Fnd the following, by using the right-hand rule:
i
j
k
,
i
k
j
,
j
i
k
,
j
k
i
,
k
i
j
,
k
j
i
,