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

Dot product geometrically:

1. Take vectors A and B.
2. Do scalar projection of A onto B. In other words take the length of the component of A that is in same direction as B (negative if opposite direction).
3. Multiply the length of B with the length of the scalar projection.

Picture: https://en.wikipedia.org/wiki/File:Dot_Product.svg

If you look at the triangles this creates you’ll see that the length of the scalar projection of A onto B is |A|cos(θ) where θ is the angle between the two vectors.

—-

Cross product geometrically:

1. Take vectors A and B.
2. Take the surface area of the parallelogram that is fromed from the vectors A and B. This surface area will be the length of the cross product.
3. Set the cross product vector to be orthogonal to both A and B according to the right hand rule (image https://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svg).

Picture: https://en.wikipedia.org/wiki/File:Cross_product_parallelogram.svg

If you again look at the triangles that form the parallelogram you will see that its surface area is |A||B||sin(θ)| where θ is again angle between A and B. (to see it easier split the parallelogram into two triangles of same size with sides formed by A and B).