In Vector Multiplications why Dot product uses cosine and Cross product uses sin ?

Question

In Vector Multiplications why Dot product uses cosine and Cross product uses sin ?

In: 2

Answer 1

Because that is how we have chosen to define them.

We could define a third kind of vector product, say an @ product, where:

> **a**@**b** = |**a**|.|**b**|.sin(θ)

or a cos version of the cross product. And those would be perfectly valid and sensible things. Except it turns out things like this aren’t quite as useful or mathematically interesting.

A lot of our vector mathematics was developed by physicists, particularly those working on electromagnetism, and it turns out that the dot and cross products are really useful in a lot of physics (the dot product relating to resolving vectors in the same direction as each other, the cross product for finding things in perpendicular directions or for doing circular motion).

Once we have got these two definitions they can be generalised more broadly into the “exterior” (or “wedge”) and “inner” products – which take the same sort of idea but apply them to more generalised vector spaces.