 Good quick mathematics of bones. I have two vectors Let me call them a And they could be Let's start with them. Just being in two space So for instance, let me this let this vector be three comma negative two for instance and this vector equals One comma two So if I draw this in a coordinate plane I have one two three one two, so very roughly. This is my vector a my vector a And my other vector is one across one two up. There's my second vector There is my second vector vector B. What is the cross product between these two vectors? So I want to know what is vector A? cross product vector B Various ways to do this. First of all, I can very easily get the direction. I point my fingers in the a direction I let them go to be my thumb right hand points in the direction of the resultant vector Let's call that vector C So I already know that it's going to the directions going to be at the board now So the direction is if this if I add a third and I've got to add a third dimension And that comes straight at you, so let's make that the z-axis. It's going to come straight out at you. Okay, I Can also I can also Well, let's do one way. The one way is to set up a matrix So the matrix is just wrote so I'm going to have this matrix i negative j Okay Put that negative there just to mind myself. You need to do that So I've got i negative j and I've got k and I now write my Three and negative two and there's nothing now We know these are the unit vectors in that direction So a length of one in the direction of the x-axis This was x and this is y and z points at you and then the z It's in this flat plane So there's no component in the z direction for that first vector for the second vector It's one two and zero so there is my three by three magics I've got three rows and three columns and all I do now I get the determinant along row one If you don't know what the determinant is the determinant gives you if you go to long row one of the three by three magics The answer to this cross product So how do you get the determinant along row one if you don't know if you can't remember? So I'm going to get three answers I'm going to have an i the j and the k component and I put that negative there just to remind myself that there must be a negative here So to get this first one it is for i a long row once I'm going to use the i the negative j and the k So for this one, I'm going to close this row and the column that contains the i What am I left with negative two and zero and this and this So what do I do I multiply those two and I multiply these two and I subtract that from each other So it's negative two times zero minus zero times two. That's zero minus zero. Okay, so if I have d e f is g It's d g multiplied minus e f Multiply those two multiply those two and it's d g minus e f So for the j I close the row that contains j column that contains j. So j is completely hidden from me So it's three times zero minus zero times one. That's also zero And then lastly I'm going to close the row and the column with k that's six minus negative two That is eight and that's a positive eight So there was a number if this was two vectors in three space And this was not zero zero I would have had numbers here. I just have to remember to make this a negative. So my answer is going to be eight positive eight okay, so a cross b The one vector times the other vector if I wanted to know what the linear angular momentum was that's r cross p Okay, I would do the cross product exactly if I knew the Units for this a x comma y units for this x comma y units for that I would set up to this matrix if it was in three space. I would hand values for x y and z I would have values for for p and x y and z and I would put it in there and I would take the determinant A long row one I would get this kind of answer which now is that equal to zero comma zero comma eight That would be the cross product of these two vectors