Dot Physics

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.

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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:

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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:

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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.

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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.

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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.

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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.

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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.

Comments

  1. #1 Rich
    July 6, 2010

    As an electrical engineering student we had to do this all the time. So, guess what the logo of IEEE is? The right hand rule.

    http://www.schajee.com/wp-content/uploads/ieee-logo.jpg

  2. #2 CalcDave
    July 6, 2010

    I just used this this morning! My wife’s car battery was dead and there’s a near-impossible bolt to reach to get the old battery out. I didn’t want to screw around (forgive the pun) trying to figure out which way to turn the ratchet, so I used the right-hand-rule to figure out which way I needed to twist to loosed the bolt!

  3. #3 Rhett Allain
    July 6, 2010

    @Rich,

    That is an awesome logo.

  4. #4 Rhett Allain
    July 6, 2010

    @CalcDave,

    Yes – I forgot to mention that normal screws and bolts follow the right hand rule. Good point.

  5. #5 Rich (civil engineer, not the electrical engineer one)
    July 6, 2010

    @Rich #1, isn’t there also something like the left-hand rule when you’re dealing with electricity? I recall something very vaguely (from 20+ years ago) about dealing with electrons/current/magnetic fields?

    And @CalcDave, I also recall that gas fittings do NOT follow the right-hand rule (a/k/a/ “righty-tighty, lefty-loosey) so that it’s more difficult to accidentally mix up gas piping with other plumbing.

  6. #6 Keith Harwood
    July 6, 2010

    Rich #5.

    IIRC the fittings for oxygen bottles have left-hand threads and most other gasses have right-hand thread. I suspect the effects of accidentally connecting a high-pressure oxygen cylinder to a high pressure hydrogen cylinder would be rather somewhat interesting.

  7. #7 JP
    July 6, 2010

    @Rich #5,

    You might be thinking about the relationship between the flow of electrons and the magnetic field. Current and magnetic field follow the right hand rule. However, the flow of electrons is opposite the direction of the current, so the relationship between electron flow and magnetic field would follow the left-hand rule, so-to-speak.

  8. #8 Fran
    July 6, 2010

    I also love watching students taking tests and using the right-hand rule! Someday I am going to take video of them…

  9. #9 Stu
    July 6, 2010

    Rich, the left hand rule is the motor rule, with your thumb/fingers pointing out the field, current and motion. Running the current through a wire in a magnetic field induces a force on the wire.

    The equivalent right hand rule is the dynamo rule, in which motion of a conductor through a magnetic field induces a current.

    Here in the UK my physics teacher taught me a mnemonic for remembering which is which:

    You drive your motor on the left, and if you drive on the right, you’ll dynamo (die-in-a-mo!). Very sad that I still remember that.

  10. #10 Lassi Hippeläinen
    July 7, 2010

    @Keith Harwood #6:
    The fittings of oxygen bottles must have no grease, because it would be a fire hazard. Using left-handed thread makes sure wrong kind of fittings won’t get mixed up accidentally.

    @Stu #9:
    Somehow most traffic in the world drives on the right and manages to survive. There must be something wrong in electricity.

  11. #11 Peeter Joot
    July 7, 2010

    I always hated the arbitrariness of the right hand rule and the cross product. It bugged me so much I went off and learned Clifford Algebra (or “Geometric Algebra”), which allows for the use of direct vector products to represent planar quantities like torque essentially as oriented planar quantities. No more right hand rule!

  12. #12 rob
    July 7, 2010

    looks around…whispers: pseudovector…runs away…

  13. #13 gary
    July 7, 2010

    you forgot the best part for the professor – watching students use their left hands “by mistake” and knowing which ones are going to fail the exam before you even start grading.

  14. #14 Cherish
    July 7, 2010

    The first time I used the right-hand rule, it made no sense…so I came up my with my own. When looking at the Lorentz force, I would align my right thumb with the velocity of the particle, align my fingers with the perpendicular component of the field (my fingers looked like field lines, so it was easy to remember), and the resulting force would be the direction my palm pushed (push = force). I know how the ‘real’ right-hand rule works, but I think my method makes more sense. :-)

  15. #15 Joseph McCullough
    November 29, 2010

    That applet link was extremely helpful, thanks. Helped me on my university physics homework assignment.

  16. #16 Rita
    February 22, 2011

    It’s really helpful. Thank you !!! ^^

  17. #17 bisso
    January 3, 2012

    how to right the moment of a couple or force in a vector form while solving a 3D problem? can i use the right hand rule in a 3D problem.

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