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

I've redrawn

Notice that this means that the direction of

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.

If we have a force

The

This torque measures the turning effect of

You can see that this must be so from this drawing looking down at the plane containing the line of action of

We have

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.

Here's another physical application of the cross product. In the drawing below, I've shown a particle of mass m with position vector

The

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

Since sin 0 = 0 and sin 90 = 1 and each vector is of unit length, we have

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

Now we can see

Suppose we have

Then **a** x **b** =
(a_{1} **i** + a_{2} **j** + a_{3} **k**)
x (b_{1} **i** + b_{2} **j** + b_{3} **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

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** = 3**i** + 2**j** + 4**k** acts through the
point with position vector **r** = 2**i** + **j** + 3**k**.

What is its torque about a perpendicular axis through O?

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

Which of the following will work?

- (2
**i .**3**i**)**.**4**j** - (2
**j .**4**j**) x 3**k** - (2
**i**x 3**j**) x 4**i** - (2
**i**x 3**j**)**.**4**k**

Which is which? How big is the box? It really is worth working out the answers to these for yourself before continuing.

or **back to the previous section**

or **back to the vectors homepage.**