One thing that we noticed last time, is that
Is it possible to prove this? In order to verify, let’s perform the exact same steps with two vectors [vx,vy] and [wx,wy]. First let’s calculate v x w:
Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [cx,cy] and [dx,dy]. For later reference, let’s identify each component:
Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:
Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:
Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:
Step 5: Place a k-hat at the end and box your answer, you’re done! So our answer is:
Next, let’s calculate -(d x c). In order to do this, we’ll simply calculate w x v and then multiply by -1.
Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [wx,wy] and [vx,vy]. Notice that they are reverse this time. For later reference, let’s identify each component:
Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:
ax * by = dx*cyStep 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:
Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:
Step 5: Place a k-hat at the end and box your answer, you’re done! So our answer multiplied by -1 is:
So, not a full proof, but very convincing nevertheless. It turns out that this is actually true in all cases, in 2-D and all higher dimensions. In order to explore the cross product deeper, we will now perform the same example, but with 3-D vectors, which will have a single new number as our z-coordinate.
