MitPY 4: Motivating Vector Products

A question from frequent commenter, Steve R:

Hi, interested to know how other teachers/tutors/academics …give their students a feel for what the scalar and vector products represent in the physical world of \boldsymbol{R^2} and \boldsymbol{R^3} respectively. One attempt explaining the difference between them is given here. The Australian curriculum gives a couple of geometric examples of the use of scalar product in a plane, around quadrilaterals, parallelograms and their diagonals .

Regards, Steve R

16 Replies to “MitPY 4: Motivating Vector Products”

  1. (c&p+edit from other thread)

    Hi Steve!

    Good question. I don’t like the first link so much because that seems to have already given up on imparting an understanding of the idea and resigned itself to teaching efficient memorisation.

    I do the dot product first and I work toward the definition using the angle and unit vectors. Then, I give the extension to non-unit vectors. I prove the more usual definition then as a computational device.

    For the cross product, I define this by first asking the general question of: given two unit vectors X and Y in \mathbb{R}^3, how can we canonically choose a third vector Z such that Z is orthogonal to X and Y? (Or if we can’t, what should we do?) Once this canonical choice is made for all X and Y, we do some examples, the extension to non-unit vectors, and of course take this to be the definition. Again, the “determinant” mnemonic (it is not a determinant!) is treated as a computational device.

    1. I’m not required to teach the cross product (so I don’t), but for the scalar product I do a similar, albeit far more terse, presentation

      Using cosine rule to find the angle between vectors \vec{a}, \vec{b} (a diagram is mandatory here), we have \left| \vec{b} - \vec{a} \right|^2 = \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - 2\left|\vec{a}\right| \left|\vec{b}\right| \cos \theta and then writing out the vectors in their i-j-k forms and doing some algebra we get \left|\vec{a}\right| \left|\vec{b}\right| \cos \theta  = a_ib_i + a_jb_j +a_kb_k.

  2. The cross product is the harder one to justify. If your students are also doing Physics, you might like to give some applications there.
    eg1: Torque = r \times F
    This is a good one as it seems natural that a torque has direction perpendicular to the plane of r and F (curl your fingers as in the twist and look at your thumb! Left-handers banned). And torques add as vectors.
    eg2: Force on a moving charge in a magnetic field
    F = q v \times B

    1. And if one is really brave… after teaching slope fields in the differential equations part of Specialist, you can talk about the curl of the vector field which is a cross-product…

      I was thinking about this as a SAC idea but gave up pretty quickly as I think it would be very tokenistic; but I’m prepared to be convinced otherwise (as always).

  3. Thanks, Glen and Tom. will try to add thoughts soon. Quickly, Glen, why is the determinant not a determinant? And, Tom, you can put LaTex in comments: just put the word “latex” (without quotes) right after the opening dollar sign, and note you can try-edit for thirty minutes after posting your comment.

    1. On determinants, there is a lovely movie called “Who killed determinants?” I showed it to students years ago.

      1. Interesting. I see the movie exists, but I don’t see that it it is easily viewable online. Obviously off-topic from Steve’s question, but the role of determinants in teaching linear algebra is obviously a tricky one to get right.

    2. On Marty’s question to Glen regarding the cross-product not being a determinant, I think Glen meant this: the cross product of two vectors \mathbf{a} = (a_1,a_2,a_3)^{\sf T} and \mathbf {b} = (b_1,b_2,b_3)^{\sf T} may be written as \mathbf{a} \times \mathbf{b}= \left|\begin{array}{ccc}  \mathbf{i} & \mathbf{j} & \mathbf{k} & \ a_1 &  a_2 & a_3 & \  b_1 & b_2 &  b_3& \end{array}\right| (where \mathbf{i}=(1,0,0)^{\sf T }, and so on), provided that vectors are allowed as entries in the determinant. As per the standard definition of a determinant, they are not.

  4. Tom,

    Torque is a good application for mechanics students. I use the example of applying a force F at angle of theta to an adjustable spanner of length r tightening a bolt and the right hand grip rule to get the direction of the cross product.

    Steve R

  5. There is a nice application of the scalar product in statistics; all students should know that the sample coefficient of correlation (r) is between -1 and +1; less well-known is that this is because r is the cosine of an angle.

    Reference
    Fitzpatrick, T., Lenard, C.T., Mills, T.M. (2013) Correlation and vectors. Vinculum, 50(2), 10.

  6. Hi everyone, thanks for the discussion, and sorry I’ve been too snowed to take part. I guess Steve is primarily asking for physical interpretations rather than mathematical definitions/properties. That’s a bit out of my world (and the school world), though I would have thought angular momentum was the most natural example of the cross product in physics.

    1. That’s interesting, because electromagnetism was the first thing that came to mind when I thought about vector products in physics. Maybe we’ve just all had different experiences.

  7. Hi ,

    thanks for all your responses and examples.

    my conclusions are that at High School level Mathematics and Physics the examination questions on dot and cross product are mainly testing the ability to apply the defining formulas. I will continue to use the physical applications of torque and work done to illustrate them

    As far as vectors in R3 the area of the parallelogram gives a reasonable geometric definition of the cross product in R3. This link illustrates https://betterexplained.com/articles/cross-product/

    Steve R

  8. Aside from everything else, I think two suggestions are worth making.

    The first is in the business of demonstrating a right-hand rule. I say don’t do the three-finger thing where you hold out three mutually perpendicular fingers on your right hand. No one I’ve ever asked can tell me quickly which finger stands for which vector. Those three fingers just stress people, and I’m not surprised. Fact is, you can make the same construction with your left hand. So, the sense of right-handedness is easily lost, and you might just have to know the right-hand rule to remember which finger is which, to give you the right-hand rule. Since no one seems to remember which finger is which, this technique just fails.

    Instead of holding out three fingers, the approach of closing all of the fingers of the right hand into the palm (of the right hand!) is a true right-handed procedure. Start with with all of your fingers pointing together in the direction of vector \boldsymbol{a}, then close them into the palm in the direction of vector \boldsymbol{b} through the smallest angle between these vectors (in their shared plane, of course). Your thumb then points in the direction of \boldsymbol{a}\times \boldsymbol{b}. This “right-hand curl” procedure is all you’ll ever need, but it can be modified to quickly produce related quantities that appear especially in electromagnetism. So that’s my first suggestion: ditch the three fingers forever as a mental aid in the right-hand rule.

    My second suggestion is that you should be aware of what happens when you consider the larger of the two angles between the vectors in their shared plane. Call the smaller of these angles \theta, and the larger 2\pi - \theta. When computing a dot product, it doesn’t matter which of these angles you use, because \cos\theta = \cos(2\pi - \theta). In fact, the same is true for the cross product, but you have to be on your toes. To compute \boldsymbol{a}\times \boldsymbol{b}, consider finding the direction of that vector by the procedure I described above, where you curl your fingers from \boldsymbol{a} to \boldsymbol{b} through \theta. Consider that this procedure gives you a unit vector, which you multiply by ab\sin\theta (which is positive) to give the actual vector \boldsymbol{a}\times \boldsymbol{b}. Notice that multiplying by a positive number didn’t change the direction of that unit vector. Now do the same thing for the angle 2\pi - \theta. Start with your fingers pointing along \boldsymbol{a}, but now curl them through 2\pi - \theta to arrive at \boldsymbol{b}. This gives you a unit vector in the opposite direction to before; but now remember to multiply it by ab\sin(2\pi -\theta)—which is negative—to give \boldsymbol{a}\times \boldsymbol{b}. Multiplying that unit vector by this negative number has switched its direction to agree with what you got a moment ago. So using the larger angle works too, but you have to remember to “multiply your thumb” by a negative number. Rather than have to remember that, it makes more sense just to close your fingers through the smaller angle \theta, and thus not to have to think about multiplying by ab\sin\text(\mbox{angle curled through}).

    The moral of the story is: for the dot product, use whichever angle you like, smaller or larger. For the cross product, use whichever angle you like, smaller or larger; but it’s a tad easier and quicker to use the smaller one.

    1. Thanks, Don. I totally agree on the right hand rule: I can never remember that damn finger thing. The larger angle observation is also nice.

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