It is that part of the semester where the Right Hand Rule (RHR) comes out. Really, the best part is the students taking the tests. They make all these funny motions with their hands. That makes tests more entertaining (for me) than they usually are.
What is the RHR?
Suppose I have two numbers. Maybe these two numbers are the length and width of a piece of paper. Now suppose I need to multiply length times width to get the area (A = L x W). Simple – right? But that is multiplication for scalar variables. How do you multiply vectors?
There are two common operations you can do with vectors. First is the dot-product (also known as the scalar product). The dot-product is an operation that takes two vectors and gives you a scalar number. This is what happens with the calculation for work.
There is also an operation that gives you a vector. This is the vector product or cross product. There are several ways to write this. First, let me just write the magnitude of vector A cross vector B:
This is not the answer. This is just the magnitude of the answer. The answer is a vector. Well, how do you find the direction of this vector? Of course, there are more sophisticated ways, but in general this resultant vector direction must:
- Be perpendicular to the vector A
- Be perpendicular to the vector B
- Follow the right hand rule.
Ok – before I do some examples, some quick notation. The problem with cross products is that they HAVE TO use 3 dimensions. This is the only way to have a resultant vector perpendicular to both of the original vectors. So, how do you draw 3D vectors on 2D paper? One notation is to use an “X” to represent a vector going into the paper (and perpendicular to the plane of the paper) and a dot to represent a vector coming out of the paper.
Here is my first example. Vector A cross vector B as shown:
The first step to find the direction is to identify the TWO vectors that are perpendicular to both the vectors A and B. There will be two. Here are two vectors that ARE NOT perpendicular to both A and B.
The two vectors that are perpendicular are a vector coming out of the paper (or screen) and going in. So, which do you choose? You choose the one such that when you put the thumb of your right hand in the direction of that vector, your right fingers cross vector A and then vector B.
Right away, you may notice that the order of operation for the cross product matters. But, here is my right hand with the thumb in the two possible directions.
In the wrong case, my fingers of my right hand would cross vector B before crossing A, so it is wrong. Maybe this is still confusing from my picture. Here is a great applet that lets you play with these vector things. Vector Cross Product Applet – Syracuse University Physics.
Now for my tips. This is really my whole point of this post. When students are studying the magnetic force (which needs the right hand rule), I tell them the following:
- If you are right handed, put down your pencil. Oh sure, this seems silly now – but when you are taking a test, you will forget. If you are left handed, don’t put down your pencil.
- Don’t hurt yourself. What does that even mean? It means be careful when doing something like this.
Seriously, if you are not careful you could get some wrist injuries.
Oh, also, I know what you are going to say. You are going to say that torque needs the right hand rule too. Well, technically you are correct. However, most intro algebra-based texts just deal with torque about a fixed axis so that you can treat it as a scalar quantity.