Cross Product
Pre-requisites: Vectors
The cross product, like the dot product,
is a product of two vectors which has two definitions.
The geometric definition of the cross product is that
[where once again
is the angle between
the two vectors]
and that the direction of the cross product is orthogonal to both
and
. From this
definition, it should be clear that the cross product of two vectors
IS A VECTOR and not a scalar. Repeat that phrase to yourself
a couple of times: the cross product is a vector. Taunt people who
believe otherwise.
The definition of the cross product I just gave you is incomplete: there
are two vectors that satisfy being orthogonal to
and
and also having
the desired magnitude. To pick the correct one, we must use the
right hand rule: if you point the thumb of your right hand
along and
point your pointing finger along
, then
will point in the direction of your middle
finger. To put it another way: if
and
are rotated
to lie in the part of the xy-plane with y >= 0, then
will point in the direction of the positive
z-axis.
The algebraic definition of the cross product is
(v_{1}, v_{2}, v_{3}) ×
(w_{1}, w_{2}, w_{3}) =
( v_{2} w_{3} - v_{3} w_{2},
v_{3} w_{1} - v_{1} w_{3},
v_{1} w_{2} - v_{2} w_{1} )
This definition is easier to remember if you think of it as the
determinant of the matrix
In fact, to some people the whole point of the cross product is that
the triple product (what you get when you combine the dot
product and the cross product) is the same as a determinant:
If you don't know what a determinant is, you can just use the above
equation as a definition and not worry about it. It's most important
property is that if you have a parallelopiped (the three dimensional
version of the parallelogram -- a solid with eight pairwise parallel
faces) with vertices
has volume
.
Thus the triple product / determinant of a three by three matrix can
be thought of as an "oriented" (signed) volume.
Now, thinking of the cross product in terms of the determinant of a
three by three matrix only helps you if you can remember the determinant
of a three by three matrix. One way of doing this is to think of it
in terms of the determinants of two by two matrices:
where
The determinant of a two by two matrix also has a geometric interpretation:
it is the area of the parallelogram with vertices
.
Wow. We've made quite a number of strong statements, without giving a hint
of proof to any of them. Let's rectify this situation:
- Statement: The algebraic and geometric definition of the
cross product are the same.
Proof: First, the algrebraic definition and the geometric
definition of the cross product agree in magnitude:
Next, the algebraic definition of
is orthogonal to both
and
:
( v_{2} w_{3} - v_{3} w_{2},
v_{3} w_{1} - v_{1} w_{3},
v_{1} w_{2} - v_{2} w_{1} )
· ( v_{1}, v_{2}, v_{3} )
= v_{2} w_{3} v_{1}
- v_{3} w_{2} v_{1}
+ v_{3} w_{1} v_{2}
- v_{1} w_{3} v_{2}
+ v_{1} w_{2} v_{3}
- v_{2} w_{1} v_{3}
= 0
and similarly for
because
.
Finally, the algebraic definition of the cross product obeys the right
hand rule because it does for
(c, 0, 0) × (a,b,0) = (0, 0, cb ) [which is "up" if (a,b,0) is on
the positive
y side of the plane, and "down" if (a,b,0) if on the negative y side of the
plane assuming that c > 0], and the general case of
can always be smoothly rotated into
that case. The right hand rule is unchanged by rotations, and the
algebraic definition of the cross product changes continuously with
smooth rotations so if it obeys it before a smooth rotation, it must
obey it afterwards too. Q.E.D.
- Statement: The magnitude of the cross product gives the
area of the parallelogram with vertices .
Proof: Trigonometry.
Corollary: The determinant of the two by two matrix
gives the area of the parallelogram with vertices (0,0), (a,b), (c,d),
and (a+c,b+d)..
Proof:
= (a,b,0) × (c,d,0)
- Statement: The absolute value of the triple product
gives the volume of the parallelopiped with vertices
.
Proof:
which
is the base of the parallelopiped times its height.
If you found the above proofs a little abstract, you may find it useful to
interactively play around with the cross product to
develop your intuitions. You may also take comfort in the fact that
the following simple properties are all that you really need to know
about the cross product:
- Anticommutativity: .
[I know that I said it already before, but it is worth repeating.]
- Linearity:
.
Also,
and
.
- Idempotence:
.
- Cyclicity:
,
,
.
Finally, here's an application of the cross product: finding the equation
of a plane given two vectors and a point lying on the plane.
We did this before by solving a system of linear equations, but
it is much simpler if you use the cross product. Just take the cross
product of the two vectors to get a vector orthogonal to both of them
and thus a normal vector to the plane, and then plug the normal vector
(A,B,C) and the point (x_{1}, x_{2}, x_{3}) into
the formula
A ( x - x_{1} ) + B ( y - x_{2} )
+ C ( z - x_{3} ) = 0
Exercises:
- What do you get when you cross a grape and an elephant?
When you cross a grape and a mountain climber?
- Is the cross product associative?
- Calculate the cross product of (1,2,3) and (4,5,6).
- Prove that the cross product is linear in its second argument.
- Prove that the cross product factors constants in its second argument.
- What is ?
?
- Find a vector orthogonal to (1,2,3) and (0,4,-9).
- Find the volume of the parallelopiped with the vertices .
- Find the equation of the plane going through the points (0,0,0),
(1,2,3), and (2,0,-1).
- Find the equation of the plane going through (1,2,3) and containing
the vectors (0,0,1) and (2,-3,7).
- Prove that the cross product of two vectors is the zero vector
if and only if the two vectors are parallel or one of them is the zero
vector.
- Prove the Jacobi Identity:
- Show that determinants can factor a scalar from a row or column.
- Show that a determinant doesn't change if one row (or column) is
added to another row (or column).
- Show that the system of equations
has a solution if and only if the determinant
isn't zero.
- Show that switching two rows (or columns) changes the sign of
a determinant.
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last modified 1 July 1997