>> When I look at this first equation, I cross J equals K and notice they're just in alphabetical order: I-J-K. Easy to remember. When I start permuting these symbols -- in other words, move this one back to this position, this one back to this position and then this one moves out, comes around front, J cross K equals I. Do it to this one. Move this one here, this one here, this one moves out, comes around front, we get K cross I equals J. Recall this sicklick [phonetic] permutation of the symbols, and it makes it very easy to remember how the basic unit vectors form their cross products. What about what's gonna happen if I reverse the order? Well, that reverses the sense. For instance, if I'm forming V cross W in this picture, it points out of the blackboard, what happens when I form W cross V? It goes into the blackboard. So changing the order in a cross product is gonna reverse the direction, but of course it can't change the area of the parallelogram, so the length stays the same. So consequently, what we have is a very simple formula for reversing the order: V cross W is the negative of W cross V. Very important to remember when it comes to cross products. Now, the cross product in fact obeys all the usual distributive and scalar laws like the dot product does, and so when I take the component form of vectors and write them in terms of I-J-K using these basic relations, I get a kind of a complicated formula for getting the cross product of vectors. And that's easy to express in terms of determinants. So in order to do this, if I wanna form V cross W, what I've done here is I've written V and W, one under the other. And here they're components. And one way to think of this is we're forming a 3 by 3 determinant by putting the basic unit vectors up here on top. But what we're doing is what's called expanding by minors; in other words, to get the I component, we eliminate this column and the row that the vectors are in, and that would tell us to concentrate on this little square here. And that gives us what's called a little 2 x 2 determinant. And to compute that, we multiply down the diagonal, A2 times B3, and then subtract, multiplying this diagonal, A3, B2. So multiply this way, and then subtract what you get multiplying by this. To get the second component, cover up the middle column and notice we have the two outer ones and we form a little 2 by 2, and in the middle we have a negative sign. And then for the final component, cover up the last one. You see we have the little 2 x 2 determinant, A1, A2, B1, B2. So in order to demonstrate this, it's actually easy. For instance, suppose we have V equals negative 3 comma 2 comma 5, and W is 2 comma negative 1 comma 3. In order to maybe help at first, you can put the unit vectors there and draw the vertical lines. Remember you're computing a determinant, in effect. V cross W is. And so what are we gonna get for the first component? Simply cover up the first column and look at 2 times 3 is 6 minus 5 times negative 1. 6 minus negative 5 is 6 plus 5, which is 11. How 'bout the second component? Cover up the middle column. Now we have negative 3 times 3 is negative 9, minus 2 times 5. That's -- [ audio breaks ] negative 10 and negative 9 is negative 19. But remember: for the middle component, we change the sign when we're through computing, so we would get plus 19. And now finally, for the last component, cover up the last column. Negative 3 times negative 1 is plus 3, minus 2 times 2 is 4. So that's 3 minus 4 is negative 1. And so there we've computed the cross product of two vectors using components.