 So yesterday we discussed the vector inner product, the inner product between two vectors. Today let's look at the other form of vector multiplication that is the cross product. Now unlike the inner product which had a resultant scalar, the vector product or the cross product is going to have another vector as its solution. You might remember from physics where you had say r cross f, the position vector. In that direction, the force vector in that direction and then by the right hand rule the torque vector was pointing in that direction. So it is another vector and no matter in which direction the two vectors point they'll always form a plane and the resultant vector will be orthogonal or perpendicular to those two. To do the cross product I have to just talk about something which we haven't discussed before and it's coming in a future lecture that's called the determinant. I just want to show you the determinant. If I have a vector 2 by 2 vector and a matrix I should say 2 by 2 matrix. So row 1 column 1, row 1 column 2, row 2 column 1, row 2 column 2. The determinant of a written like that is this going to be, it's got to be a square matrix. And for 2 by 2 it's easy as it gets 3 by 3 it gets a lot harder. We just do these diagonals. So it's going to be the product a11 times this one a21 minus the product of these two. Don't get the order the wrong way around. a12 and a21. So if I were to have a matrix and that matrix was 3, 2, 4, 1 the determinant of that matrix A is going to be 3 minus 8 that's negative 5. That way multiplication minus this way multiplication. So just remember that's the determinant and we are going to discuss the determinant coming up. So let's have two vectors. I have my vector v and let's make that 3 and 2 and I have my vector w and my vector w is say 2 and 2. So if I were just to look at it 1, 2, 3, x and 1, 2 so that one's going to be 3, 2 that's not going to be my vector v and 2, 2 it's going to be my vector w. So right hand rule v and then w the resultant cross product is going to point straight at you. Look if I do w times v it points in that direction these do not commute. This product of a vector does not commute. These two lie in the plane so it's going to come out in this direction. There's one more thing that I just want to remind you of we spoke about it and that is the unit vectors. If I have three axes mutually perpendicular let's make that i, let's make that x let's make that y and let's make that z and if I let these little unit vectors go along these and they are mutually orthogonal we usually call this i hat, j hat and k hat or e sub x or e sub i and e sub j there's various notations for this doesn't matter it is these unit vectors so this vector here if I put another 0 there and another 0 there it's still the same exactly the same that I can 3 in the i hat direction so 3 times this plus 2 times that it's still going to be this very same vector v here and then 0 in the z direction so even if I had drawn it in a different way so that z comes out here towards you it's still going to be 0 on this z direction and the way that we're going to do this vector multiplication as far as cross products are concerned we're going to have an i hat, j hat and k hat and my first vector was 3 in the i hat direction 2 in the j hat direction, 0 in the k hat direction and it was going to be 2 2 0 for that nice little 3 by 3 matrix and I'm going to use this idea of determinants I'm not going to get the determinant of this 3 by 3 matrix but I'm going to use the determinant one thing that I just want to remember is plus minus plus and if it carried on to a high dimension plus minus plus minus you'll see now because I want to do something in the i hat direction and that's always going to be plus then there's going to be minus something in the j hat direction and then plus something in the k hat direction and what I do to get this i hat direction is I'm going to close off the row that contains i and the column that contains i and I'm left with this little 2 by 2 matrix and I get it's determinant 0 minus 0 is 0 so for the j hat I remember the minus that must be out there it's that row in that column 3 times 0 and 0 times 2 so 0 minus 0 0 and then for the k I close off the k and I close off the column of the k and I'm left with this little 2 by 2 matrix so I have 6 minus 4 is 2 so v cross w is going to equal 0 0 2 there we go in that order and if I were to swap those two around I'm not going to get the same I'm not going to get the same solution if it's w cross v it's going to be 0 0 minus 2 it's going to be into the board so any kind of vector that you do I mean hopefully in your exams you'll only get these multiplication of vectors that are in 3 dimensions so as I say if this was a 4 and that was a 1 that would be a 4 that would be a 1 and I would have some values there but that's the easy way to do a vector cross product remember to put those extra 0's in if they just lie in 2 dimensions but remember they're actually in 3 dimensional space there's going to be 1 coming out the result at least remember that they do not commute and remember this easy way of doing it if you have to do it with pencil and paper let's go to Math America and I'll show you how easy it is to do the cross product over there so let's look at the cross product between 2 vectors let's use the proper function which is cross so I'm just going to do the cross product and I'm going to pass my 2 vectors let's keep it in 2 dimensional space 3.2 but remember the solution is going to be in 3 directions so I've got to put that extra 0 there and the other one is 2.2 and I put my extra 0 for the z axis and I close that and indeed I see 0 0 2 so that is going to represent this orthogonal vector let's just do that in the other direction let's just make sure that we find at least one example to show that it doesn't commute and then we know the cross product of vectors do not commute so it's 2.2, 0 for the one and the other one is 3.2, 0 and there we go and we see the solution the negative 2 so same magnitude but pointing in the opposite direction so the vector cross product does not commute now just to show you there is another way to go about this I'm just going to have 3, 2, 0 and then I can hold down the escape key and then right cross and escape key again and I get this tiny little cross that is not the same as multiplication so it's not the star 8 the shift 8 to get the star that is different so 2.2, 0 there we go and I get the same solution 0, 0, 2 so create your vectors it's got to be in the same space and you've got to add the extra 0 because whatever vector you create it's going to be orthogonal to that and you can go up to much higher dimensional space as well so very easy in Mathematica to get a vector cross product