Dot Product

$a\cdot b=a_x{b}_x+a_y{b}_y=|a||b|\cos \theta .$

Where the cross product measures turning, the dot product measures alignment.

long long dot(const P& a, const P& b) { return a.x * b.x + a.y * b.y; }
long long dot(const P& o, const P& a, const P& b) { return dot(a - o, b - o); }
long long norm2(const P& a) { return dot(a, a); }        // squared length

What the sign means

$a\cdot b\{\begin{array}{ll}>0& \text{angle}<90°\text{(same general direction)}\\ =0& \text{perpendicular}\\ <0& \text{angle}>90°\end{array}$

Task	Test
Is the angle at $b$ in triangle $abc$ obtuse?	$\mathrm{dot}(b,a,c)<0$
Is the triangle right-angled?	some $\mathrm{dot}=0$
Is point $p$ within the “slab” of segment $ab$?	$\mathrm{dot}(a,b,p)\ge 0$ and $\mathrm{dot}(b,a,p)\ge 0$
Are two lines perpendicular?	direction dot $=0$
Is $p$ on ray $ab$ (given collinear)?	$\mathrm{dot}(b-a,p-a)\ge 0$

Projection

${\mathrm{proj}}_b(a)=\frac{a\cdot b}{|b{|}^2}b,\text{scalar projection}=\frac{a\cdot b}{|b|}.$

// closest point on the LINE through a,b to point p
double t = (double)dot(p - a, b - a) / norm2(b - a);
P closest = a + (b - a) * t;
 
// closest point on the SEGMENT: clamp
t = max(0.0, min(1.0, t));

The clamp is the whole difference between line and segment. Distance from a point to a segment:

double distToSegment(P p, P a, P b) {
    if (a == b) return abs_(p - a);
    double t = (double)dot(p - a, b - a) / norm2(b - a);
    t = max(0.0, min(1.0, t));
    return abs_(p - (a + (b - a) * t));
}

Squared distance — use it

long long dist2(P a, P b) { return norm2(a - b); }

Exact in integers, no square root, and order-preserving. Use it for:

• comparing distances,
• “is this within radius $r$”: $d^2\le r^2$,
• nearest-neighbour searches,
• closest pair.

Only take the square root when the problem asks for an actual length in the output.

Law of cosines

$|c{|}^2=|a{|}^2+|b{|}^2-2(a\cdot b),$
which is where the dot product’s geometric meaning comes from. Rearranged, it gives the angle:

double angle(P a, P b) {            // angle between vectors, in [0, pi]
    return atan2(fabs((double)cross(a,b)), (double)dot(a,b));
}

Using atan2(|cross|, dot) is more numerically stable than acos(dot / (|a||b|)), which loses precision for angles near 0 and $\pi$ and can feed acos a value slightly outside $[-1,1]$. Prefer the atan2 form.

Dot and cross together

Want	Use
Angle magnitude	$atan2(|\mathrm{cross}|,\mathrm{dot})$
Signed angle	$atan2(\mathrm{cross},\mathrm{dot})$ — gives $(-\pi ,\pi ]$
Perpendicular?	$\mathrm{dot}=0$
Parallel?	$\mathrm{cross}=0$
Which side?	sign of $\mathrm{cross}$
In front or behind?	sign of $\mathrm{dot}$

The pair $(\mathrm{cross},\mathrm{dot})$ is exactly the complex number $\bar{a}b$, which is why complex arithmetic works so smoothly for 2D geometry — see Vectors.

3D

The dot product is unchanged: $a\cdot b=a_x{b}_x+a_y{b}_y+a_z{b}_z$. It still gives perpendicularity, projection and the angle. Only the cross product changes character (from scalar to vector) when moving to 3D.

See also: Cross Product · Vectors · Distances