 Now, the cross-product is itself an important bilinear map for vectors on a vector space, F3. I'm going to keep this in the context of R3 because some of these statements don't make sense for every vector space, but some things we can say are the following. If you take the cross-product of u and v and you dot it with u, you end up with zero. This is to say that u is orthogonal to u cross v. It's also true here. If you take v dot u cross v, you're going to get zero as well. The cross-product is orthogonal to the original vectors. If we go back to the example we saw beforehand, if you took u to be the vector 1, 2, negative 2, you take v to be the vector 3, 0, 1, we calculated that cross-product u cross v to be the vector 2, negative 7, negative 6. If we compute those dot products u dot u cross v, we end up with 2 minus 14 plus 12. That's a zero. Likewise, we take v dot u cross v. That would end up to be 6 plus 0 minus 6, which is likewise zero. Sure enough, we can see that these vectors are perpendicular. That is the cross-product is perpendicular to the original two vectors. We're going to talk about this a little bit more in just a second, but there are some huge applications to this from a geometric point of view. Imagine we're in R3, which is the setting in which we can use the cross-product. We have a plane right here. Let's say that we know the spanners of this plane. We have some vector u and we have some other vector v like so. The cross-product has the property that it is orthogonal to the plane. This is u cross v. This gives us what we've talked about before as a normal vector. It's a vector which is orthogonal to everything in that plane. There are some benefits of having that orthogonal vector right there. The cross-product exactly gives us one of these orthogonal vectors. There are some other properties of the cross-product like to mention here. If you take the length of the cross-product squared, you can compute that using lengths of the vectors in the dot-product. You also get a law of sines going on here with the cross-product. Similar to the law of cosines we had seen previously, the length of the cross-product is equal to the length of u times the length of v times sine of the angle between them. This fact right here is used in the proof of the previous result about calculating the area of the parallelogram in R3. The next result is actually one of the most interesting right here, both parts e and f right here. You'll notice that if you take u cross v cross w, you get one thing. If you take u cross v cross w, you get another thing. You'll notice the first vector is the same u dot w times v, u dot w times v. That's the same. You'll notice these parts are different, the u dot v w versus the v dot w u. That is the people who are in the dot-product and whose screen scale is switched around. What actually happens in general is that these two things are not typically equal to each other. This is a bizarre thing, but this cross-product is what you would call a non-associative operation. That is, doing parentheses in different ways gives you a different outcome. It's also true that for the cross-product is non-commutative. The order for which you put these things makes a big difference. Now, admittedly, we've seen this with matrix multiplication. Matrix multiplication is non-commutative. In some respect, we're used to that, but a non-associative operation, we haven't really seen that too much at all yet. If you take the cross-product with the zero vector, you end up with the zero vector. This comes from the fact that the cross-product is a distributive operation. It distributes across addition. If you have a scalar k right here, if you take k times the vector u cross v, you could scale the first factor, you could scale the second factor. I want you to mention that if you put these things together, this shows that the cross-product is bilinear. It's linear in the first factor, it's linear in the second factor. And if it's a bilinear function, then if you times by zero, you'll always get zero since it's the additive identity. But it's non-commutative bilinear operation, and it's also non-associative, which is kind of weird. And then there's also this other interesting fact right here. If you take the product of a vector with itself, the cross-product, you get back zero. There are some interesting linear algebra properties right here. So I just kind of want to mention to you.