Mnemonics: Cross Product
There are two ways to multiply vectors. One of these is the “cross product”. The cross product always yields a third vector perpendicular to both of the two vectors, whose magnitude is equal to the area of the parallelogram defined by the two vectors. The cross product is defined as:
A x B = [ a3b2 + a2b3 , a1b3 + a3b1 , a2b1 + a1b2 ]
Where A and B are vectors in 3-space.
There are three difficult to remember things about the cross product:
The direction of the resulting vector.
How to calculate the numerical value of the cross product from the two vectors’ components.
And what it means.
Mnemonic #1:
If in 3-space, positive k points forward (away from you), your first vector is your index finger, your second vector is your thumb, and your resulting vector is your middle finger. In this case, remember that when admonishing demons, you hold the CROSS in your RIGHT hand as well as that you point your MIDDLE FINGER at other people, AWAY FROM YOU.
If in 3-space, positive k points backward (towards you), your first vector is your index finger, your second vector is your middle finger, and your resulting vector is your thumb. In this case, also remember that when admonishing demons, you hold the CROSS in your RIGHT hand but that you do a THUMBs up TOWARDS YOU when you succeed.
Mnemonic #2
Above, you can see how to calculate the cross product. To remember the pattern, know this:
The third child (the baby) always comes first, the first child (first born) always comes second, but the second child (middle child) always comes last. So, you remember, 3 1 2. Placing each of these in our vector, we get:
[ 3 , 1 , 2 ]
Next remember that what comes next in each is what is not there and not the position it is in. So in the first one, there is a three and it is first, so what goes there is 2. Doing this for all three positions yields:
[ 3 2 , 1 3 , 2 1 ]
Next, take what you have, reverse it, and append it so that you have three palindromes.
[ 3 2 2 3, 1 3 3 1 , 2 1 1 2 ]
Next place a’s and b’s alternating before the numbers.
[ a3 b2 a2 b3 , a1 b3 a3 b1 , a2 b1 a1 b2 ]
Now, place an additional symbol (+) in the middle of each section.
[ a3 b2 + a2 b3 , a1 b3 + a3 b1 , a2 b1 + a1 b2 ]
In the remaining spots for operations, place a multiplication symbol (*).
[ a3 * b2 + a2 * b3 , a1 * b3 + a3 * b1 , a2 * b1 + a1 * b2 ]
If, negative k points away from you, you need to place a minus symbol (-) at the beginning of each section.
[ -a3 * b2 + a2 * b3 , -a1 * b3 + a3 * b1 , -a2 * b1 + a1 * b2 ]
And that’s the cross product. So, 5 simple patterns will yield the complex equation for the cross product.