The dot product: angles, projection, facing

The dot product of two vectors multiplies matching components and adds the results. The answer is a single number, not a vector:

a · b = a.x·b.x + a.y·b.y + a.z·b.z

It is cheap (three multiplications and two additions), and it has a second, geometric meaning that makes it useful everywhere:

a · b = |a| |b| cos θ, where θ is the angle between a and b.

The sign alone says a lot

You do not even need the lengths for this: the lengths are always positive, so the sign of a · b is the sign of cos θ. That makes “is this face pointing towards the camera?” or “is this wall perpendicular to that one?” a one-line check.

The angle between two directions

Rearranging gives the angle: θ = acos(a · b ÷ (|a| |b|)). The result is always between 0° and 180°: the dot product does not know which way you turned, only how far. The cross product, in the next lesson, supplies the direction.

1using V2 = (double X, double Y);2 3static double Dot(V2 a, V2 b) => a.X * b.X + a.Y * b.Y;4static double Len(V2 v) => Math.Sqrt(Dot(v, v));5 6static double AngleDeg(V2 a, V2 b)7{8    double c = Dot(a, b) / (Len(a) * Len(b));9    c = Math.Clamp(c, -1, 1);          // rounding can give 1.000000000000000210    return Math.Acos(c) * 180 / Math.PI;11}12 13V2 wall = (4, 0);14foreach (V2 other in new V2[] { (3, 3), (0, 5), (-2, -2) })15    Console.WriteLine($"Dot = {Dot(wall, other)}, angle = {AngleDeg(wall, other):F1} deg");

Dot = 12, angle = 45.0 deg
Dot = 0, angle = 90.0 deg
Dot = -8, angle = 135.0 deg

Projection: how far along, how far off

If u is a unit vector, v · u is the length of v’s “shadow” on u’s line: how far v goes in u’s direction. That is the widget’s green arrow. In BIM terms, with u along a wall, it answers “how far along the wall is this point?”, which is how tools measure a door’s position, a pipe’s station or a dimension along a run.

Use the wall’s perpendicular instead (u turned 90°: (−u.y, u.x)) and the same dot product gives the signed distance off the wall line. Together, the two numbers are the point’s coordinates in the wall’s own system: a first taste of local coordinates.

1using V2 = (double X, double Y);2 3static double Dot(V2 a, V2 b) => a.X * b.X + a.Y * b.Y;4 5V2 a = (1, 1), b = (9, 7);                  // wall location line6V2 p = (6, 6);                              // a column centre7 8V2 ab = (b.X - a.X, b.Y - a.Y);             // (8, 6)9double len = Math.Sqrt(Dot(ab, ab));        // 1010V2 u = (ab.X / len, ab.Y / len);            // along the wall11V2 n = (-u.Y, u.X);                         // across it, to the left12V2 ap = (p.X - a.X, p.Y - a.Y);13 14double along = Dot(ap, u);15double across = Dot(ap, n);16V2 foot = (a.X + u.X * along, a.Y + u.Y * along);17 18Console.WriteLine($"Along the wall: {along:F2} m of {len} m");19Console.WriteLine($"Off the line: {across:F2} m (positive = left side)");20Console.WriteLine($"Nearest point on the line: ({foot.X:F2}, {foot.Y:F2})");

Along the wall: 7.00 m of 10 m
Off the line: 1.00 m (positive = left side)
Nearest point on the line: (6.60, 5.20)

If along is below 0 or above the wall’s length, the column lies beyond one end. The Lines and curves module turns this into a full closest-point routine for segments.

Facing: which way does a face point?

Every face has a normal: a unit vector pointing straight out of it. Its dot product with another direction tells you how much the face points that way. With unit vectors, the dot product is cos θ, so it doubles as a “how much” score from −1 to 1. Solar studies, back-face culling in viewers and “which façade does this window belong to?” all use it.

1using V2 = (double X, double Y);2 3static double Dot(V2 a, V2 b) => a.X * b.X + a.Y * b.Y;4 5V2 south = (0, -1);6var windows = new (string Name, V2 Normal)[]7{8    ("W1", (0, -1)), ("W2", (0.6, -0.8)), ("W3", (0.8, -0.6)), ("W4", (1, 0)), ("W5", (0, 1))9};10double limit = Math.Cos(45 * Math.PI / 180);      // within 45 deg of south11 12foreach (var w in windows)13{14    double d = Dot(w.Normal, south);15    string verdict = d >= limit ? "faces south" : d > 0 ? "partly south" : "no south sun";16    Console.WriteLine($"{w.Name}: dot = {d:F2} -> {verdict}");17}

W1: dot = 1.00 -> faces south
W2: dot = 0.80 -> faces south
W3: dot = 0.60 -> partly south
W4: dot = 0.00 -> no south sun
W5: dot = -1.00 -> no south sun

1using V2 = (double X, double Y);2 3static double Dot(V2 a, V2 b) => a.X * b.X + a.Y * b.Y;4static V2 Unit(V2 v) { double l = Math.Sqrt(Dot(v, v)); return (v.X / l, v.Y / l); }5 6static string Relation(V2 a, V2 b)7{8    const double tol = 1e-9;9    double c = Dot(Unit(a), Unit(b));10    if (Math.Abs(c) < tol) return "perpendicular";11    if (Math.Abs(Math.Abs(c) - 1) < tol) return "parallel";12    return $"at {Math.Acos(Math.Clamp(c, -1, 1)) * 180 / Math.PI:F1} deg";13}14 15Console.WriteLine(Relation((4, 0), (0, 3)));16Console.WriteLine(Relation((2, 2), (-3, -3)));17Console.WriteLine(Relation((1, 0), (1, 1)));18Console.WriteLine(Relation((3, 4), (-4, 3)));

perpendicular
parallel
at 45.0 deg
perpendicular