in the last video I explained how to calculate determinants in this video as promised I'm going to explain why they're good for vectors so we'll start by giving a way of calculating the cross product of two vectors from coordinates of course we had ways of calculating the dot product and adding vectors in coordinates it would be nice to have a similar way of calculating the cross product there is one and it uses a determinant so here's the rule so if we have two vectors one with coordinates a b and c and the other with coordinates x y and Zed we can calculate the cross product and the cross product is given by the determinant of a matrix whose top row consists of the vectors i j and k whose middle row consists of the coordinates of U and whose bottom row consists of the coordinates of V that's the rule Let's do an example suppose we have vectors U given by 2 I minus J plus K and V equal i - 4 J + 2 K so I had to write these down I couldn't remember what ones to use so the cross product is the determinant of the Matrix i j k 2 -1 1 1 - 4 two so let's calculate that determinant well we get I times the determinant of what's left which is -1 1 - 4 2 minus J times the determinant of what's left when you remove its row and column which is 2 2 1 1 plus K times the determinant of what's left which is 2 - 1 1 - 4 well this determinant is -1 * 2 which is -2 -1 * -4 so that's -2 - -4 so that's 2 I then we subtract 4 * 1 so 4 -1 * J so that's - 3 J and then we add on 2 * -4 - -1 * 1 so that's - 8 + 1 which is - 7 K so that's how to work out cross products of two vectors if we have the coordinates we have to work out a determinant so there some arithmetic to be done but often it's a lot easier than going through the definition of the cross product where we have to use Trigon ometry we have to use the sign of the angle between them to get the magnitude and we have to work out a vector perpendicular to both u and v to work out the direction so if we have the coordinates working out the cross product by doing this is often much easier [Music] now we know how to work out cross products in coordinates and we've known how to work out dot products in coordinates for some time we do know how to work out scaler triple products in coordinates of course scaler triple product just consists of a DOT and across so we can do it bit by bit but there's a quick way of working out the Scala triple product using a matrix so suppose we have three vectors U equals a i + BJ + C K and V equal Pi I + QJ + R K and W = X I + yj+ z k then it's a fact the rule is that the scalar triple product U do V cross W that's also given by a determinant and this time the Matrix whose determinant we take the three rows are just all the coordinates of these three vectors in order so the rule is we take the determinant of the Matrix a b c p QR x y z and then working out the determinance of that Matrix is pretty quick let's work an example again I've written down the numbers I'd like to take so suppose our Vector U is 3 I Min - J + k v is I + 4 J - 2 K and W is 3 i - 2 J + 5 K then we're looking for the determinant of the Matrix where we just take all the coord coordinates so we're looking for the determinant of the Matrix 3 -1 1 1 4 - 2 and 3 that's not three two minus 2 five let's work out that determinant so that determinant equals 3 * the determinant of what's left namely 4 - 2 3 uh - 2 5- -1 * the determinant of 1 - 2 3 5 what left when we remove this column + 1 * the determinant of 1 4 3 - 2 which is what we get when we remove the top on the right hand row and column that's equal to 3 * 4 * 5 is 20 - - 2 * -2 -4 - - 1 * something which is same same plus 1 * 5- - 2 * 3 that's + 6 + 1 * 1 * - 2 is -2 - 4 * 3 means -12 okay so this is 16 so we have 3 * 16 is 48 plus 5 + 6 means + 11- - 2 - 12 so that's -4 the answer is 48 + 11 -4 which is 45 so we worked out the scalar triple product of three vectors given their coordinates using a 3X3 determinant and that wasn't too difficult certainly much easier than working out the vector product and the dot product one at a time using the definition only