Calculus II - Cross Product
Anyhow, the point of all this you're comfortable with basic linear algebra, especially with how the determinant behaves under elementary row and column . In mathematics and vector algebra, the cross product or vector product is a binary operation on . Conversely, a dot product a ⋅ b involves multiplications between As explained below, the cross product can be expressed in the form of a determinant of a special the above given relationship can be rewritten as follows. Volume - but your link is easiest to consider for only a few vectors A matrix represents a linear transformation applied to a set of 'axes' that in turn can represent.
Should the cross product difference between interacting vectors be a single number too? Sine is the percentage difference, so we could use: Should the dot product be turned into a vector too? Well, we have the inputs and a similarity percentage. Geometric Interpretation Two vectors determine a plane, and the cross product points in a direction different from both source: If you hold your first two fingers like the diagram shows, your thumb will point in the direction of the cross product.
I make sure the orientation is correct by sweeping my first finger from vec a to vec b. So, without a formula, you should be able to calculate: Again, this is because x cross y is positive z in a right-handed coordinate system. I used unit vectors, but we could scale the terms: Calculating The Cross Product A single vector can be decomposed into its 3 orthogonal parts: Similar to the gradientwhere each axis casts a vote for the direction of greatest increase.
The final combination is: Example Time Again, we should do simple cross products in our head: Cross Product In this final section of this chapter we will look at the cross product of two vectors. We should note that the cross product requires both of the vectors to be three dimensional vectors. Also, before getting into how to compute these we should point out a major difference between dot products and cross products. The result of a dot product is a number and the result of a cross product is a vector!
Be careful not to confuse the two. There are two ways to derive this formula. Both of them use the fact that the cross product is really the determinant of a 3x3 matrix.
The first method uses the Method of Cofactors. Here is the formula. First, the terms alternate in sign and notice that the 2x2 is missing the column below the standard basis vector that multiplies it as well as the row of standard basis vectors. This method says to take the determinant as listed above and then copy the first two columns onto the end as shown below. So this term right here, we're just going to have to do our expansion of the square of a binomial. And we've done this multiple times.
So this is going to be equal to a2 squared b3 squared. And then we're going to have these two multiplied by each other twice. I'm just multiplying this out. Minus 2 times a2 a3 b2 b3.
The cross-product and determinants
I'm just rearranging them to get the order right. Plus a3 squared b2 squared. Then I have to add this term. So plus a3 squared b1 squared minus 2 times both of these terms multiplied. Minus 2 times a1 a3 b1 b3. Plus that term squared.
And then finally, this term squared. So plus a1 squared b2 squared minus 2 times a1 a2 b1 b2.
Plus a2 squared b1 squared. So there you go.
And let's see if we can write this in a form-- well, I'm going to write this in a form that I know will be useful later. So what I'm going to do is I'm going to factor out the a2, a1, a3 squared terms. So I could write this as-- let me pick a new neutral color. So this is equal to, if I just write a1 squared, where's my a1 squared terms? I got that one right there and I have that one right there. So a1 squared times b2 squared plus b3 squared.
This would be 3 squared. Now where are my a2 squared terms? So times b1 squared. And then finally, let me pick another new color. I can go back to yellow. Plus a3 squared times-- well that's that term and that term.
So b1 and b2. So b1 squared plus b2 squared. And obviously, I can't forget about all of that mess that I have in the middle, all of this stuff right here. So plus, or maybe I should write minus 2 times all of this stuff. Let me just write it real fast. So it's a2 a3 b2 b3 plus a1 a3 b1 b3. Plus a1 a2 b1 b2. Now let's put this aside for a little bit. Let me put this on the side for a little bit.
We'll let that equation rest for a little while. And remember, this is just an expansion of the length of a cross b squared. That's all this is. So just remember that. And now, let's do another equally hairy and cumbersome computation.
Proof: Relationship between cross product and sin of angle
Let's take this result up here. We know that the magnitude or the length of a times the length of b times the angle between them is equal to a dot b. Which is the same thing as if we actually do the dot product, a1 times b1 plus a2 times b2 plus a3 times b3.
Now, just to kind of make sure that I get to do the hairiest problem possible, let's take the square of both sides. If we square this side, you get a squared b squared cosine squared. Then you got a dot b squared or you get this whole thing squared. So what's this whole thing squared? For me, it's easier to just write out the thing again. Instead of writing a square, just multiply that times a1 b1 plus a2 b2 plus a3 b3.
And let's do some polynomial multiplication. So first, let's multiply this guy times each of those guys. So you have a1 b1 times-- well there a1 b1. I'm going to do it right here.
You get a1 squared b1 squared plus a plus this guy times this guy. Plus a1 a2 times b1 b2. Plus this guy times that guy. Plus a1 a3 times b1 b3. Now let's do the second term. We have to multiply this guy times each of those guys. So a2 b2 times a1 b1. Well that's this one right here.
I wrote it right here because this is really the same term and eventually, we want to simplify that. So that's that times that guy. Then we have this guy times that over there. So let me write it over here. So that's a2 squared b2 squared. Put a plus right there. And then finally, this middle guy times this third guy.
Plus-- so a2 a3 b2 b3. Now, we only have one left. Maybe I'll do it in this blue color.