Like the dot product, the vector cross product is another means of vector multiplication. The difference with the cross product being that the operation yields a vector quantity. To be honest, I had some hesitation including this in the bootcamp. It’s one of those things I honestly don’t think you’ll use incredibly often. However, if needed, it’s a very handy tool, and if you’re in a class where you do need it, it’s best to have seen it at least once before. So, without further ado, here are the 5 steps to performing the cross product operation in 2 Dimensions:
1) Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]
2) Multiply ax and by
3) Multiply ay and bx
4) Subtract step 2 from step 3, i.e. ax*by-ay*bx
5) Place a k-hat at the end and box your answer, you’re done!
What the heck k-hat? In 2-dimensions, the unit vectors which point in the “x” and “y” directions are called i-hat and j-hat. k-hat points along the z-direction in a 3-D coordinate system and has a length of 1 unit. This gives the vector its direction. When given two vectors, the cross product vector will always point perpendicular to both vectors. More on this shortly. But, it is possible to write your cross product vector as [0,0,ax*by-ay*bx]. So, the cross product kind of forces you to be in 3 dimensions whether you like it or not.
So if that’s the case, how does it work in 3 dimensions? Different steps, and 6 steps this time:
1) Identify Two Vectors in Coordinate System as [ax,ay,az] & [bx,by,bz]
2) Calculate [ay*bz-by*az]*i-hat
3) Calculate [az*bx-ax*bz]*j-hat
4) Calculate [ax*by-ay*bx]*k-hat
5) Write your vector as a vector consisting of the last 3 steps: [step2,step3,step4]
6) Box your answer, you’re done!
So, these are the steps to follow when calculating a cross product in 2 or 3 dimensions. Actually performing examples will make this process clearer and more fluid.
