Let and compute the cross product of
In terms of the standard basis vectors you can check the multiplication
table. An easy way to remember
the multiplication table is to put the vectors clockwise in a circle . Given two of the three
vectors their product is either plus or minus the remaining vector. To determine the sign you step from
the first vector to the second, to the third: if this makes you go clockwise you have a plus sign, if you
have to go counterclockwise, you get a minus.
The products of and
are all you need to know to compute the
cross product. Given two vectors
and write them as and, and multiply as follows
This is a useful way of remembering how to compute the
cross product, particularly when many of the
components and are zero.
44.4 Example. Compute
There is another way of remembering how to find
×. It involves the
“triple product” and
determinants. See § 44.3.
44.2. Algebraic properties of the cross product
Unlike the dot product, the cross product of two vectors behaves much less like ordinary multiplication.
To begin with, the product is not commutative – instead one has
for all vectors and.
This property is sometimes called “anti-commutative.”
Since the crossproduct of two vectors is again a vector you can compute
the cross product of three
vectors You now have a choice: do you first multiply and , or and , or and ? With
numbers it makes no difference (e.g. 2× (3 × 5) = 2 ×15 = 30 and (2 × 3) ×5 = 6 ×5 = also 30) but
with the cross product of vectors it does matter: the cross product is not associative, i.e.
for most vectors
so"×” is not associative
holds for all vectors,
and any number t. We were already using these properties when we multiplied
in the previous section.
Finally, the cross product is only defined for space vectors, not for plane vectors.
44.3. The triple product and determinants
Definition 44.5. The triple product of three given vectors,,
is defined to be
In terms of the components of ,, and one has
This quantity is called a determinant, and is written as follows
There’s a useful shortcut for computing such a determinant:
after writing the determinant, append a fourth and a fifth column
which are just copies of the first two columns of the determinant.
The determinant then is the sum of six products,
one for each dotted line in the drawing . Each term has a sign:
if the factors are read from top-left to bottom-right, the term is
positive, if they are read from top-right to bottom left the term
is negative .
This shortcut is also very useful for computing the cross
product. To compute the cross product of two
given vectors and you arrange their components in the following determinant
This is not a normal determinant since some of its entries
are vectors, but if you ignore that odd circumstance
and simply compute the determinant according to the definition (67), you get (68).
An important property of the triple product is that it is much more symmetric in the factors ,,
than the notation suggests.
Theorem 44.6. For any triple of vectors one has
In other words, if you exchange two factors in the product
it changes its sign. If you
“rotate the factors,” i.e. if you replace by, by and by , the product doesn’t change at all.
44.4. Geometric description of the cross product
Proof. We use the triple product:
since for any
vector . It
is perpendicular to.
Proof. Bruce just slipped us a piece of paper with the following formula on it:
diligently computing both sides we find that this formula
actually holds for any pair of vectors ,! The (long) computation which implies this identity will be
presented in class (maybe).
If we assume that Bruce’s identity holds then we get
since . The theorem is
These two theorems almost allow you to construct the cross product of two vectors geometrically. If
and are two vectors, then their cross product satisfies the following description:
(1) If and are parallel, then the angle θ between them vanishes, and so their cross product is
the zero vector . Assume from here on that and are not parallel.
is perpendicular to both
and . In other words,
are not parallel, they
determine a plane, and their cross product is a vector perpendicular to this plane.
(3) the length of the cross product× is sin θ.
There are only two vectors that satisfy conditions 2 and 3: to determine
which one of these is
the cross product you must apply the Right Hand Rule (screwdriver rule, corkscrew rule, etc.) for
,,×: if you turn a screw whose axis is perpendicular to and in the direction from to , the
screw moves in the direction of ×.
45. A few applications of the cross product
45.1. Area of a parallelogram
Let ABCD be a parallelogram. Its area is given by
“height times base ,” a formula which should be
familiar from high school geometry.
If the angle between the sides AB and AD is θ, then the height of the parallelogram is ,
so that the area of ABCD is
The area of the triangle ABD is of course half as much,
These formulae are valid even when the points A, B, C, and D are points in space. Of course they
must lie in one plane for otherwise ABCD couldn’t be a parallelogram.