Angles

In this lesson we stop asking which side of a line something is on and ask for the angle itself.

Two ways to write the same angle

The dot product has an angle hidden inside it:

and the cross product hides the same angle, with its sine instead:

Divide the second by the first and both lengths cancel, leaving $\tan \theta$. Undoing that is exactly what atan2 is for, and because it takes the two parts separately it also gets the quadrant right:

The answer comes back in $(-\pi ,\pi ]$ and it is signed - positive when b is counter-clockwise from a, the same rule the orientation gave us.
To convert it to degrees we multiply it by: 180.0 / acos(-1.0);

Implementation

In simpler terms, we have a function that when given the cross and dot product, returns an angle.

#include <bits/stdc++.h>

using namespace std;

struct v{

    long long x, y;
};

long long cross(v a, v b){

    return (a.x * b.y) - (a.y * b.x);
}

long long dot(v a, v b){

    return (a.x * b.x) + (a.y * b.y);
}

double angle(v a, v b){ //signed angle from a to b, in degrees

    return atan2((double)cross(a, b), (double)dot(a, b)) * 180.0 / acos(-1.0);
}

int main(){

    cout<<fixed<<setprecision(1);

    cout<<angle({1, 0}, {0, 1})<<'\n';
    cout<<angle({1, 0}, {-1, 0})<<'\n';
    cout<<angle({0, 1}, {1, 0})<<'\n';
    cout<<angle({3, 0}, {3, 3})<<'\n';

    return 0;
}

Output:
90.0
180.0
-90.0
45.0

This is the first place in the chapter where we leave the integers. When a problem only asks which of two angles is bigger, the cross product answers that exactly - reach for atan2 when the angle itself is part of the answer.