What does a determinant have to do with a cross product? pt. 2

In the previous post, I showed that the quantity



and the determinant




are both tests that the three vectors , , and are independent. Now we'll go about showing that they're actually the same test - that they're the same number.

You may be familiar with the geometric interpretation of as the volume of a parallelipiped (box) with sides , , and .

Just a quick explanation - is a vector perpendicular to both, whose length is the area of the parallelogram they form. See stolen image below:



If I take that and dot it with , I get the projection of into the perpendicular direction, times the area of the box. That is, I get the height of the box times the area of its base. That's the volume. Wikipedia visualization:














There's one subtlety, which is that the volume of the box records its "handedness" in the sign of . By handedness, I mean that if we looked at a mirror image of the box, its volume would be multiplied by negative one. If you do that funny thing with your right hand, and find that points the same sort of direction that does, you have a righthanded box. Otherwise it's left-handed.

This interpretation of as the volume turns out to be the same as the determinant. To see why, we'll make a list of properties that the area of the box has, and then show that those properties imply the normal calculation of the determinant.

• If the sides of the box are orthonormal (all perpendicular to each other, and all unit length), its volume is one (or minus one for a left-handed box).
  
• The volume of the box is linear in a vector. By this I mean that if we multiplied any side of the box by some constant, we'd multiply the volume of the entire box by that constant. Also, if we make two boxes that share two vectors but differ in the third, the sum of the volumes is the same as the volume of the box whose third side is the sum of the vectors. Notationally:

and similarly for and

• If you switch the roles of any two vectors, you'll multiply the volume of the box by minus one. This is obvious if you flip and , because then the angle between them gets measured the opposite way, so its sine is the opposite of what it used to be. The other possibilities you can visualize if you just play around with it a little. It works.

With these properties, we can uniquely determine the value of the volume of the box from its components, and show that it's the determinant. It's a little long for three dimensions, so you can work through that on your own if you want, after seeing it in two dimensions.

Imagine we have two 2-D vectors in component form, and we put them in a matrix. Then we ask for the determinant. The first property says the determinant of the identity is one.




Using linearity in the vectors, we find





Now we switch the vectors in the identity and multiply the determinant by -1.



and



Finally use the linearity within a vector, which states



where we could drop the determinants of matrices with an entire column zero because they are dependent vectors.

That is the determinant for a 2-by-2 matrix. If you want to do it for 3-by-3, you can use the exact same process of breaking it down into 27 matrices with three nonzero entries each, and then see by permutations of the identity which ones are zero, and where the plus and minus signs go. Out will fall the normal expression for a 3-by-3 determinant.

So that's where that comes from.

Why it Matters

In the previous post, we were trying to solve the problem of taking three vectors and finding a linear combination of them that yielded a fourth vector. That is, find , , such that



where , , , and are all arbitary vectors.

We worked far enough just to test if the problem had a solution, but we found this result:



From there, we need to find by dotting it with a unit vector perpendicular to both and .



Plug this into the equation for , and the stuff about cancels, leaving



Those are determinants. That's why they matter. They let you solve systems of linear equations, such as the system of linear equations that results from the problem we just solved.