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// SPDX-License-Identifier: CC-BY-SA-4.0
// Copyright © Paweł Redman
#import "/common.typ": post, link-self, static, img
#show: post.with(
title: "Angle between two vectors",
date: datetime(year: 2020, month: 4, day: 17),
updated: datetime(year: 2026, month: 6, day: 14),
)
#figure(
img(static("math/angle_between_vectors.png")),
caption: [Which one to choose?],
)
In $RR^n$ there are two angles between any two vectors. The smallest one I'll call $theta$. The other one, its 360° complement, I'll call $phi$.
The smallest angle between two vectors $bold(u)$ and $bold(v)$ is given by:
$
theta = cos^(-1)((bold(u) dot bold(v)) / (u v))
$
which follows easily from the geometric definition of the dot product:
$
bold(u) dot bold(v) = u v cos theta
$
The other angle is simply the 360° complement:
$
phi = 2 pi - theta
$
= Oriented angle
#figure(
img(static("math/oriented_angles_between_vectors.png")),
caption: [Oriented angles between example vectors *u* and *v*.],
)
Some applications call for the *oriented* angle $psi$. It's the angle going from one vector to the other in a specific direction. The convention that I use is that the direction be *counter-clockwise*.
Because the angle is oriented, changing the order of vectors (or the used convention) changes the angle's sign:
$
psi(bold(u), bold(v)) = -psi(bold(v), bold(u))
$
or, equivalently:
$
psi(bold(u), bold(v)) = 2 pi - psi(bold(v), bold(u))
$
Furthermore:
$
psi(bold(u), bold(u)) = 0
$
It's also worth noting that:
$
cos theta = cos phi = cos psi
$
which is why the dot product doesn't carry enough information to compute $psi$.
== In two dimensions
$
psi(bold(u), bold(v)) &= op("atan2")(bold(u) times_2 bold(v), bold(u) dot bold(v))
$
where the "two-dimensional cross product" is defined as:
$
bold(u) times_2 bold(v) = u_1 v_2 - u_2 v_1 = u v sin theta
$
== Derivation: in two dimensions
Using the geometric definitions of the two-dimensional cross product (above) and the dot product:
$
cases(
bold(u) dot bold(v) = u v cos theta,
bold(u) times_2 bold(v) = u v sin theta,
)
$
Solving for the sine and cosine:
$
cases(
cos theta = 1 / (u v) bold(u) dot bold(v),
sin theta = 1 / (u v) bold(u) times_2 bold(v),
)
$
Solving for $theta$ (see #link-self("math/atan2#systems")[Systems of equations involving the cosine and the sine of an unknown]):
$
theta &= op("atan2")(1 / (u v) bold(u) times_2 bold(v), 1 / (u v) bold(u) dot bold(v)) \
&= op("atan2")(bold(u) times_2 bold(v), bold(u) dot bold(v))
$
The simplification can be done because $1 / (u v) > 0$ and eliminating it will not change the result (see #link-self("math/atan2")[properties of atan2]).
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