The cross product

If someone hands us to scalars -- simple numbers, like 5 and 3 -- and says "multiply these!", there is no ambiguity. There's only one way to multiply two scalars.

On the other hand, if we are told to multiply two VECTORS,

then there are two possibilities:

• the dot product takes two vectors as inputs and yields a single SCALAR as the output.

  
    
              Q:  What is the dot product of vectors A and B
                           shown above?
  
                  (Not sure what to do?   Maybe this will help. )
  
         

  My answer.

• the cross product of two vectors produces as output a third VECTOR, which is always perpendicular to the two input vectors.

Let's learn more about the cross product. It's a complicated subject, so let's take it a little at a time.

First, this product of two vectors always produces a vector which is perpendicular to the two inputs. So, for example, if we are given two vectors in the x-y plane,

then their cross product must be in the +z or -z direction.

How do we decide between the +z and -z possibilities? The answer is -- surprise, surprise -- to use the right-hand rule.

1. raise your right hand
2. point your fingers in the direction of the first vector ("A")
3. curl your fingers (without breaking them) towards the second vector ("B")
4. your thumb points in the direction of the cross product

So, in the example above, A × B points in the POSITIVE z direction.

On the other hand, the cross product B × A produces a result in the NEGATIVE z direction.

Let's practice. I'll pick two vectors, and you tell me the direction of their cross product.

Okay, that rule tells us the direction of the cross product. What about its magnitude?

Well, there are two ways to compute it. If you are given the angle θ between the two input vectors

then the magnitude of the cross product is

On the other hand, if you are given the input vectors in component form, like this,

then you need to use some algebra. One can combine the components of the two vectors to form the cross product A x B like so:

Let's try! Look again at the example we saw earlier:

  Q:  What is the cross product of vectors A and B?

One way to remember this particular arrangement is to consider the cross product as a determinant:

Now, when we compute the torque in a physics problem, we are interested in one particular cross product:

Note several things:

• the maximum size of the torque is the product of the magnitude of r and the magnitude of F
• the direction of the torque will be perpendicular to both r and F
• if a force points straight toward (or away from) the axis of rotation, so that the angle θ is zero or 180 degrees, then the torque due to that force is zero
• the order of the arguments matters! If you reverse the r and F, you will end up with a torque in the wrong direction!

Example of the cross product

Given two vectors:

1. What is the cross product A × B?
2. What is the angle between these two vectors?