The cross product: normals, area, left or right

The dot product turns two vectors into a number. The cross product turns two 3D vectors into a third vector:

a × b = (a.y·b.z − a.z·b.y, a.z·b.x − a.x·b.z, a.x·b.y − a.y·b.x)

The formula is easiest to remember by its pattern: for the X part, use the Y and Z parts, and so on round the cycle X → Y → Z → X. What matters more is what the result means:

• Direction: a × b is perpendicular to both a and b, so it is perpendicular to the plane they lie in. That is exactly what a face normal is.
• Which way: it follows the right-hand rule. Fingers along a, curl towards b, and your thumb points along a × b. So x × y = z in a right-handed system.
• Length: |a × b| = |a| |b| sin θ, the area of the parallelogram spanned by a and b. Half of that is the area of the triangle.
• Order matters: b × a = −(a × b). Swapping the two flips the normal.
• Parallel vectors give zero: sin 0° = 0, so a × b = 0 is the standard test for parallel directions.

1using V3 = (double X, double Y, double Z);2 3static V3 Cross(V3 a, V3 b) =>4    (a.Y * b.Z - a.Z * b.Y, a.Z * b.X - a.X * b.Z, a.X * b.Y - a.Y * b.X);5 6Console.WriteLine($"X x Y = {Cross((1, 0, 0), (0, 1, 0))}");7Console.WriteLine($"Y x Z = {Cross((0, 1, 0), (0, 0, 1))}");8Console.WriteLine($"Y x X = {Cross((0, 1, 0), (1, 0, 0))}");

X x Y = (0, 0, 1)
Y x Z = (1, 0, 0)
Y x X = (0, 0, -1)

In plan: one number, two answers

For two vectors lying flat in the XY plane, the X and Y parts of the cross product are zero and only the Z part is left: a.x·b.y − a.y·b.x. This number is often called the 2D cross product or the perp-dot product, and it is one of the most used expressions in plan geometry.

• Its sign gives the turn: positive when b is counter-clockwise from a (to its left), negative when clockwise (to its right), zero when they are parallel.
• Its size gives the area: the parallelogram’s area, or twice the triangle’s.
• Side of a wall: with a = B − A (the wall) and b = P − A, the sign says whether P is left or right of the wall as you walk from A to B. Divide by |B − A| and you get the distance from the line too.

1using V2 = (double X, double Y);2 3static double Cross(V2 a, V2 b) => a.X * b.Y - a.Y * b.X;4 5V2 a = (0, 0), b = (10, 0);                 // wall from A to B, walking east6V2[] points = { (4, 2), (4, -1.5), (12, 0) };7 8foreach (V2 p in points)9{10    double c = Cross((b.X - a.X, b.Y - a.Y), (p.X - a.X, p.Y - a.Y));11    string side = c > 1e-9 ? "left" : c < -1e-9 ? "right" : "on the line";12    Console.WriteLine($"{p}: cross = {c}, {side}");13}

(4, 2): cross = 20, left
(4, -1.5): cross = -15, right
(12, 0): cross = 0, on the line

In 3D: the normal of a face

Take any three corners of a flat face, P0, P1 and P2. The two edges P1 − P0 and P2 − P0 lie in the face, so their cross product points straight out of it. Normalise it and you have the unit normal; take half its length and you have the triangle’s area. Here is a sloping roof face:

1using V3 = (double X, double Y, double Z);2 3static V3 Sub(V3 a, V3 b) => (a.X - b.X, a.Y - b.Y, a.Z - b.Z);4static V3 Cross(V3 a, V3 b) =>5    (a.Y * b.Z - a.Z * b.Y, a.Z * b.X - a.X * b.Z, a.X * b.Y - a.Y * b.X);6static double Length(V3 v) => Math.Sqrt(v.X * v.X + v.Y * v.Y + v.Z * v.Z);7 8V3 p0 = (0, 0, 0), p1 = (4, 0, 0), p2 = (0, 4, 3);   // eaves corners and a ridge corner9V3 n = Cross(Sub(p1, p0), Sub(p2, p0));10double len = Length(n);11V3 unit = (n.X / len, n.Y / len, n.Z / len);12 13Console.WriteLine($"Cross: {n}");14Console.WriteLine($"Unit normal: {unit}");15Console.WriteLine($"Triangle area: {len / 2} square metres");16Console.WriteLine($"Roof slope: {Math.Acos(unit.Z) * 180 / Math.PI:F2} deg");

Cross: (0, -12, 16)
Unit normal: (0, -0.6, 0.8)
Triangle area: 10 square metres
Roof slope: 36.87 deg

The normal points up and towards −Y, so this roof face looks south (with Y as north): exactly what the dot-product solar check in the previous lesson needs. And the angle between the normal and straight up (0, 0, 1) is the roof’s slope: since both are unit vectors, their dot product is just unit.Z.

Building a frame from one direction

Two cross products turn a single wall direction into a full set of axes for the wall: side = dir × up points to the wall’s right, and up = side × dir recovers a true vertical. This “frame from a direction” pattern is how tools orient doors, windows and section boxes, and you will use it in the Matrices module.

1using V3 = (double X, double Y, double Z);2 3static V3 Sub(V3 a, V3 b) => (a.X - b.X, a.Y - b.Y, a.Z - b.Z);4static V3 Cross(V3 a, V3 b) =>5    (a.Y * b.Z - a.Z * b.Y, a.Z * b.X - a.X * b.Z, a.X * b.Y - a.Y * b.X);6static double Length(V3 v) => Math.Sqrt(v.X * v.X + v.Y * v.Y + v.Z * v.Z);7static double TriangleArea(V3 a, V3 b, V3 c) => Length(Cross(Sub(b, a), Sub(c, a))) / 2;8 9var roof = new (V3 A, V3 B, V3 C)[]10{11    ((0, 0, 0), (4, 0, 0), (0, 4, 3)),12    ((4, 0, 0), (4, 4, 3), (0, 4, 3))13};14 15double total = 0;16foreach (var t in roof) total += TriangleArea(t.A, t.B, t.C);17Console.WriteLine($"Roof area: {total} square metres");

Roof area: 20 square metres