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Let $a$, $b$, $c$ be vectors in $\mathbb{R}^3$ that form a triangle.
(How this even makes sense formally, I don't know, since in a vector space all vectors are "glued" to the origin. In an affine space, or in $\mathbb{R}^3$ viewed as a manifold, it'd make more sense, perhaps.)

Let $a \bullet b$ be the dot product of vectors $a$ and $b$.

Let $a \times b$ be the cross product of vectors $a$ and $b$.

Let $|a|$ be the norm of vector $a$.

Now:

$$\cos[a,b] = {a \bullet b \over |a|\cdot|b|}$$

and

$$\sin[a,b] = {|a \times b| \over |a|\cdot|b|}$$

(Note. This last equation can't be 100% true because the number |a| is nonnegative for all vectors $a$.)

The real number $|a \times b|$ is just the norm of the cross product $a \times b$.

Which leads to the question:

The real number $a \bullet b$ is the norm of what vector product?

Also: can you replace the cross product $\times$ with the wedge product $\wedge$ to generalize the sine equation to higher dimensions?

étale-cohomology
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    $|a|^2=a•a$, what do you mean when you say $a•b$ is a norm? – Paul Aug 08 '23 at 08:25
  • Are you looking for this? https://en.wikipedia.org/wiki/Polarization_identity – Travis Willse Aug 08 '23 at 08:33
  • As regards, your second question, the short answer is yes. For the case of the angle between $2$-planes (so, in dim. $\geq 4$), see: https://math.stackexchange.com/questions/910920/angle-between-two-planes-in-four-dimensions/911775#911775 – Travis Willse Aug 08 '23 at 08:36
  • @Paul I mean to say that $a \bullet b$ it a real number, and so (modulo sign) there exists a vector (indeed, infinitely many, lying in a circle) for which it is the norm. I'm wondering which of those vectors would have a "nice" interpretation. And what kind of vector product (if any) yields that vector. – étale-cohomology Aug 08 '23 at 18:27
  • @TravisWillse Thanks for that, but I'm looking for the sine between (1D) vectors in higher dimensions (just like we have the cosine), not for planes (or hyperplanes). Still, that's a nice answer. – étale-cohomology Aug 08 '23 at 18:35
  • In short the formula generalizes to arbitrary dimension. For vectors it is $|{\mathbf a} \wedge {\mathbf b}| = |{\mathbf a}| |{\mathbf b}| |\sin \theta|$, which you can verify with a coordinate computation. – Travis Willse Aug 08 '23 at 23:04
  • One case when the roots of an irreducible quintic are expressible in radicals is when the Galois group is the Frobenius group, $F_{20}$: https://math.stackexchange.com/questions/3136962/how-to-calculate-the-galois-group-of-x515x12 – Travis Willse Aug 08 '23 at 23:24
  • What is $\cos[a]$? What is the cosine of a vector? – jjagmath Nov 16 '23 at 23:15
  • You're looking at the definition of what that cosine is. (But it had a typo; now should be fixed.) – étale-cohomology Nov 17 '23 at 03:22

1 Answers1

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I think I may have it (at least in $\mathbb{R}^2$). Praise the Lord Jesus Christ.

The dot product $a \bullet b$ of vectors $a,b \in \mathbb{R}^2$ is:

  • the norm of the vector projection of vector $|b| \cdot a$ to vector $b$,
  • the norm of the vector projection of vector $|a| \cdot b$ to vector $a$,

where $s \cdot a$ is the scalar product when $s$ is a scaler and $a$ is a vector.

In other words,

  • extend $a$ to a circle of radius $|a| \cdot |b|$, and then project towards (the line defined by) $b$, or
  • extend $b$ to a circle of radius $|b| \cdot |a|$, and then project towards (the line defined by) $a$.
étale-cohomology
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  • I don't think you’ve answered the question you posed. I also don’t know why you introduced the first paragraph and $c$. Assuming $a\cdot b\ge 0$, it is the norm of the cross product $Ja\times b$, where $J$ is rotation through $\pi/2$ in the plane spanned by $a$ and $b$. – Ted Shifrin Nov 17 '23 at 04:54