B Why is sine not used for dot product?

AI Thread Summary
Cosine is used for the dot product because it represents the simplest metric for determining the relationship between two vectors, specifically their projection onto each other. The dot product, defined as A·B = |A||B|cos(angle), calculates how much of one vector lies in the direction of another. Using sine would complicate this by introducing additional information that would need to be accounted for. The discussion also touches on the geometric interpretation of cosine as the starting point of the unit circle, where cos(0) = 1 and sin(0) = 0, emphasizing its role in vector projections. Overall, the dot product's reliance on cosine stems from its straightforward application in measuring vector alignment.
Kirkkh
Messages
1
Reaction score
0
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).
 
Mathematics news on Phys.org
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.
 
  • Like
Likes WWGD
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)
 
Kirkkh said:
“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.
Kirkkh said:
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.
 
  • Like
Likes jedishrfu
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Back
Top