Proof relationship between cross product and sin of angle

Vector Calculus: Understanding the Cross Product – BetterExplained

proof relationship between cross product and sin of angle

Proof: Relationship between the cross product and sin of angle between vectors. Proof that $\left\| \vecb{u} \times \vecb{v} \right\| = \left\| \vecb{u} \right\|\left\| \vecb{v } \right\|\sin\theta \quad $ for $ \vecb{u} \not= \vecb{0}, \quad \vecb{v} \not. (Remember that trig functions are percentages.) Should the cross product ( difference between interacting vectors) be a single number too? is proportional to the magnitude of each vector and the “difference percentage” (sine). If you like, there is an algebraic proof, that the formula is both orthogonal and of size ‖ a.

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.

  • Proof: Relationship between cross product and sin of angle

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.

Proof that ||u x v|| = ||u|| ||v|| sin(theta) for u,v not equal to 0

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: Crossing the other way gives -vec k. I did z first because it uses x and y, the first two terms. Now the y component: There are theoretical reasons why the cross product as an orthogonal vector is only available in 0, 1, 3 or 7 dimensions.

However, the cross product as a single number is essentially the determinant a signed area, volume, or hypervolume as a scalar. Connection with Curl Curl measures the twisting force a vector field applies to a point, and is measured with a vector perpendicular to the surface. Instead of multiplication, the interaction is taking a partial derivative. As before, the vec i component of curl is based on the vectors and derivatives in the vec j and vec k directions.

Relation to the Pythagorean Theorem The cross and dot product are like the orthogonal sides of a triangle: I cheated a bit in the grid diagram, as we have to track the squared magnitudes as done in the Pythagorean Theorem. Next, remember what the cross product is doing: 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.

proof relationship between cross product and sin of angle

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. 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.

proof relationship between cross product and sin of angle

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.

Vector Calculus: Understanding the Cross Product

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. I have to multiply this guy times each of those guys. So a3 b3 times a1 b1. That's the same thing as this term right here. Because you have a3. Let me write it right here. You have a3 b3 times a1 b1. Then you have this guy times that guy, which is this because it's a3 b3 times a2 b2.

Let me put a little plus sign there. And then finally, you have this guy times himself. So you have a3 squared b3 squared. And so if you add up all of this business here what do you get?

I'll switch to another color. You have a1 squared b1 squared. Plus, and I'm doing these colors in a certain way on purpose, plus a2 squared b2 squared. Plus a3 squared b3 squared. Plus, and let me do it in this, I'll do it in white. Plus, what do you have here? You have this term times 2. And then you have this term times 2.

proof relationship between cross product and sin of angle

So plus 2 times a let me write that down. Plus 2 times a1 a2 b1 b that's that term. Plus this one right here. Plus a1 a3 b1 b3.

proof relationship between cross product and sin of angle

Finally, plus this one. And you might have noticed something interesting already. If you compare this term right here, if you compare that guy right there to this guy right there, they're the same thing. You have an a1 a2 b1 b2, a1 a2 b1 b2. This term and that term are the same. Let's look at the other terms. Let me pick a nice color. That term and that term is the same. And then finally, if you compare a2 a3 b2 b3.

This shouldn't be a plus. This is just this one. So a2 a they're all multiplied. This term and this term is the same. And in this expression, when we expanded it out, we have 2 times this, positive 2 times this. And this term right here when we expanded it out, we have minus 2 times this.

So let's see if we can simplify things a little bit. So what happens if we add this guy to this guy? So it's a little exciting. So we get a cross b, the length of that squared. And we're going to add to that this expression right here, so plus the length of a squared times the length of b squared times the cosine of the angle between them squared.

What's that going to equal? It's going to be equal to this thing plus this thing. Let's do a simplification. What's this thing plus this thing? Well we already said this is the minus 2 times this. This is the plus 2 times this. So this guy-- let me be very clear.

proof relationship between cross product and sin of angle

This right here is going to cancel out.