Geometric Interpretation of the Cross Product

There is as well a geometric interpretation of the cross product.  Firstly we will let θ be the angle in between the two vectors a^→ and b^→ and suppose that 0 ≤ θ ≤ Π, and after that we have the subsequent fact,

||a^→ *|| = ||a^→|| ||b^→|| sin θ

and the following figure.

There must be a natural question at this point. How did we be familiar with that the cross product pointed in the direction that we have given it hereθ

First, as this diagram, implies the cross product is orthogonal to both of the original vectors. This will all time be the case with one exception that we'll obtain to in a second. 2^nd, we knew that it pointed in the upward direction (in this case) through the "right hand rule".

This says that if we take our right hand, begin at a^→ and rotate our fingers towards b^→ our thumb will point towards the cross product. Hence, if we'd sketched in b^→ * a^→ would have gotten a vector in the downward direction.