Professor Dave here, let’s find cross products.
Earlier in the series, we learned about vectors and their operations, including one way to
multiply two vectors together, which we called the dot product.
But the dot product is just a scalar.
There is another way to multiply two vectors to get another vector, and this new vector
is called the cross product, which is denoted by the multiplication sign.
Now that we know a little bit about matrices and their operations, we are ready to understand
how to find the cross product of two vectors, so let’s learn how to do that now.
Let’s take two vectors, A and B. A has components one, three, four, which can also be expressed
as I plus three J plus four K, and B has components two, seven, negative five, which can be expressed
as two I plus seven J minus five K.
The way we find this cross product, or A cross B, is by finding the determinant of the following
three by three matrix, with I, J, and K across the top row, the components from vector A
in the second row, and the components from vector B in the third row.
We know how to get the determinant of a three by three matrix, so this is no problem.
We just get I times the determinant of this two by two matrix, minus J times the determinant
of this one, plus K times the determinant of this one.
Now we can find each of these determinants.
First we have negative fifteen minus twenty eight, then negative five minus eight, and
then seven minus six.
Simplifying and combining with the unit vectors, we are left with negative forty three I, plus
thirteen J, plus K.
So we can see how using determinants, we have found the cross product of these two vectors,
which is another, different vector.
We must understand that a vector cross product will always yield a vector that is orthogonal,
or perpendicular, to the original two vectors, or rather to the plane containing them.
In fact, if you place the edge of your right hand directly on vector B and curl your fingers
in the direction of vector A, so that your hand is sort of mimicking the angle the angle
formed between them, your thumb will be pointing in the direction of their cross product.
Alternately, you can point your index finger in the direction of A, and your middle finger
in the direction of B, and again your thumb will be pointed in the direction of the cross product.
So the right-hand rule that we originally learned for three-dimensional coordinate systems
can help us understand the direction of a cross product.
There are a few other key points to understand about cross products.
First, if you take the cross product of a vector and itself, like A cross A, you will
always get zero.
This will be easy to prove to yourself if you work out the determinant just as we did
before and see that all the terms cancel out.
Also, we can make the following statement about the length of the cross product vector.
The magnitude of A cross B is equal to the magnitude of A times the magnitude of B times
the sine of the angle between them.
This definition can also be used to find the cross product of two vectors, because magnitude
and direction are the two pieces of information conveyed by any vector, and if we calculate
the magnitude of the cross product this way, and then determine its direction by using
the right-hand rule, we can therefore accurately define this cross product.
From this method we can also arrive at another truth, that any two parallel vectors must
have a cross product equal to zero.
That’s because parallel vectors have an angle of zero between them, and the sine of
zero is zero, so the whole cross product, according to this expression, must go to zero.
One other interesting thing about the cross product is that the magnitude of the resulting
vector, which is the AB sine theta expression we just used, will also be equal to the area
of the parallelogram created by vectors A and B.
This is another reason that the cross product of parallel vectors is the zero vector, because
they span zero area.
Lastly, just as we learned some properties of the dot product, let’s quickly mention
some properties of the cross product.
First, the cross product is not commutative.
A cross B does not equal B cross A. Rather, A cross B equals negative B cross A. Second,
the cross product is not associative.
The quantity A cross B, cross C is not equal to A cross the quantity B cross C. However,
the cross product does distribute.
A cross the quantity B plus C is equal to A cross B plus A cross C.
This understanding of the cross product will suffice for our purposes, and we should note
that the cross product crops up all the time in physics, but let’s keep going with linear
algebra, right after we check comprehension.