Why is sine not used for dot product?

[Kirkkh]

TL;DR Summary

Re-examine an old question on here from 2012 “Why sine is used for cross product and cosine for dot product?

There’s a old 2012 post on here “Why sine is used for cross product and cosine for dot product?” —there are a lot of great answers (which is how I came about this forum). After reading over the replies, it occurred to me: really it’s just because cosine is the “start” of a unit circle.

Which is to say we set up a “dot product” to be a single number, it’s a simple idea —how do two similar vectors relate? Given that —we use the simplest metric (cosine). If we used sine (again, not being at the bringing of the circle) it would add additional information that would need to be subtracted.

So I guess a better question would of been: why is cosine the beginning of the unit circle? (which I’m sure there’s many good reasons for).

 

[marcusl]

It is because the dot product is the projection of one vector onto another. If you draw a diagram and calculate the projected length of the shorter onto the longer vector, it goes as cosine of the angle by elementary trigonometry. Invoking a unit circle would restrict dot products to unit vectors.

 

[jedishrfu]

I look at it in a vector sense. I have a two vectors A and B and I want to know what is the projection of A on B and what portion of A is NOT projected on B ie is perpendicular to B.

The projection can be found by the dot product A.B = |A||B|cos(angle_between_A_and_B).

This comes from thinking of A as the hypotenuse of a right triangle and B as a side and |B|/|A|= cos(angle_between_A_and_B) . Notice |A|sin(angle_between_A_and_B) is the other side of the triangle.

Hence, for the projection of A on B we get |A|cos(angle_between_A_and_B)

Next for the portion of A that is NOT projected onto B ie that is perpendicular to B.

We use the cross product and sin(angle_between_A_and_B) and for symmetry we define the result as a vector perpendicular to both A and B:

whose length is: |AxB| = |A||B|sin(angle_between_A_and_B)

 

[Svein]

Check out inner product and Hilbert space (https://en.wikipedia.org/wiki/Hilbert_space).

 

[Mark44]

“Why sine is used for cross product and cosine for dot product?” —there are a lot of great answers (which is how I came about this forum). After reading over the replies, it occurred to me: really it’s just because cosine is the “start” of a unit circle.

??
Why do you think that cosine is the "start" of the unit circle?
The "start" of the unit circle would be the point (1, 0) for an angle of 0 (radians). At this point $\cos(0) = 1$ and $\sin(0) = 0$. So both trig functions are involved, as they are at all points of the unit circle.

Given that —we use the simplest metric (cosine). If we used sine (again, not being at the bringing of the circle) it would add additional information that would need to be subtracted.

Why do you think that cosine is the simpler metric?
In relation to two vectors with components among the reals, there are two definitions for the dot product: a coordinate definition, and a coordinate-free definition.
The coordinate definition is $u \cdot v = u_1v_1 + u_2v_2 + \dots + u_nv_n$.
The coordinate-free definition is $u \cdot v = |u||v|\cos(\theta)$, where $\theta$ is the angle between the two vectors.