(8) The vector or cross product

Is it possible to find a way to multiply two vectors together so that we get a vector result?
The problem is that, working from the two directions of our starting vectors, we have to find some way of using these to give us the single direction of the resulting vector.
This problem makes it impossible to define this kind of vector multiplication in 2 dimensions but there is a very neat way of solving it in 3 dimensions.
We first slide the two vectors together, so that their tails meet. (As long as at least one of them is a free vector, we will always be able to do this.) Now they both lie in one particular flat surface or plane. We can use the direction perpendicular to this plane to give us the direction of our vector product. We only have to decide which of the two possible perpendicular directions to choose.
The drawing below shows how we do this to find the vector product of the two vectors a and b.
I've redrawn b shifted so that the tails of a and b meet.
how the vector product works
We now define the vector product of a and b as
a x b = |a||b| sin t n
where n is a unit vector perpendicular to the plane in which a and b lie. (I've shown n as part of the vector a x b.) We choose the direction of n so that a, b and n can fit along the thumb, first finger and second finger of your right hand. (Check this with your hand and my drawing.)
Notice that this means that the direction of b x a is given by - n so a x b = - b x a.
The multiplication sign used to show the vector product is called 'cross'. It is also sometimes written as a little upside down v like a chinese hat.
(If you are visiting from finding a normal vector, you can return here.)
This definition has some very neat practical applications. The drawing below shows one of these.
how the vector product gives torque

If we have a force F acting through a point P with position vector r with respect to O, then F and r lie in a plane through O. I have also redrawn the force vector F shifted so that its tail is at O.
The torque or moment of F about an axis through O perpendicular to this plane is given by
T = r x F = |r||f| sin t n

This torque measures the turning effect of F about the axis through O. It is independent of the position of P on the line of action of F.
You can see that this must be so from this drawing looking down at the plane containing the line of action of F and the origin, showing the position vectors of 3 different choices of point on this line of action.
any point on the line of action of F gives the same torque about O

We have |p| sin P = |q| = |r| sin R so the torque of F about the perpendicular axis through O is independent of the point you choose.

My next drawing shows that the value of sin t does indeed give the various practical possibilities, when a person applies torque using a lever to turn on a tap.

3 possible ways of turning a lever
Here's another physical application of the cross product. In the drawing below, I've shown a particle of mass m with position vector r relative to an origin O. If this particle has velocity v then its momentum p is given by p = mv.
The angular momentum or moment of momentum of the particle is defined as
angular momentum = r x p = r x mv = m(r x v).


how the vector product gives angular momentum

So not only does this definition of vector multiplication make sense mathematically but, as I've shown, it also has extremely useful physical meanings.

Some special cases

What answers do we get if we work out the cross products for the different possible pairs we can choose from the unit vectors i, j and k?
these unit vectors and the right-hand rule

Since sin 0 = 0 and sin 90 = 1 and each vector is of unit length, we have
i x i = j x j = k x k = O, (the zero vector).

Also, i x j = k and j x k = i and k x i = j

while j x i = - k and k x j = - i and i x k = - j.

You can see how the various plus and minus signs come by using the right-hand rule for each product.
I've shown this for the case i x j = k.
Now we can see how vector multiplication will work out using components.
Suppose we have a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k.

Then a x b = (a1 i + a2 j + a3 k) x (b1 i + b2 j + b3 k).

It can be shown that it is all right to work this out by working out all the 9 separate little cross products that we get from multiplying these two brackets together. Doing this, and using the results above, we get

a x b = (a2b3 - a3b2) i + (a3b1 - a1b3) j + (a1b2 - a2b1) k

and this gives us the rule for how we work out a cross product using components.

Here is a numerical example using this result.
A force F = 3i + 2j + 4k acts through the point with position vector r = 2i + j + 3k.
What is its torque about a perpendicular axis through O?

The torque = r x F = (1x4 - 3x2) i + (3x3 - 2x4) j + (2x2 - 1x3) k = - 2i + j + k.

This method is very much easier than trying to work out the moment of a force in 3 dimensions using geometry.
(If you are visiting from can we multiply 3 vectors together? or finding a normal vector, you can use these links to return.)

Thinking point

Can we also multiply vectors in threes? (We'll need to know which two we should start with.)
Which of the following will work? Two are impossible. One gives a vector answer and one gives the volume of a neat little box.
Which is which? How big is the box? It really is worth working out the answers to these for yourself before continuing.


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