GameWarp

This query was initially titled:

What’s the relation between the scalar product and dot product of two vectors?

However that did not match what the physique of the query asks, so let’s begin by clarifying some phrases:

The scalar product and the dot product are two names for the similar factor, written $vec a cdot vec b$.

What’s written on the left facet of the equation there’s the magnitude of the vector product, additionally known as the cross product.

With that terminology cleared up, we are able to say one thing concerning the equation in query:

$$start{align}
left | vec a instances vec b proper| &= vec a cdot vec b
ab left| sin theta proper| &= ab left| cos theta proper|
finish{align}$$

The place $a ge 0$ and $b ge 0$ are the lengths of vectors $vec a$ and $vec b$, respectively, and $theta$ is the angle between them.

We will see that this equation will maintain true if both $a = 0$ or $b = 0$ (both $vec a$ or $vec b$ or each are the zero vector), or if the angle obeys $ left| sin theta proper| = left| cos theta proper|$, which occurs if there’s an angle of $frac {pi} 4$, $frac {- pi} 4$, $frac {4 pi} 4$, or $frac {-3 pi} 4$ between them.

(The latter two are the identical separation between the vectors as the primary two, simply “within the reverse order”. That is the distinction between $vec a$ being clockwise or counterclockwise of $vec b$, in case you repair one view/orientation of the aircraft containing them. This issues for the signal of the cross product, however not its magnitude: $ vec a instances vec b = – vec b instances vec a$)