Adding, subtracting and scaling vectors

Vectors have just three basic operations, and all three work component by component: do the same thing to the X parts, the Y parts and the Z parts. That is all the arithmetic you need for most of this course.

Adding: chaining steps

A pipe run is a chain of straight runs. Each run is a vector, and the pipe’s end is its start plus every run added up. The order does not matter for where you end up (a + b = b + a), which the widget’s parallelogram shows: both routes round it meet at the same corner.

1using V3 = (double X, double Y, double Z);2 3static V3 Add(V3 a, V3 b) => (a.X + b.X, a.Y + b.Y, a.Z + b.Z);4 5V3[] runs = { (3, 0, 0), (0, 2, 0), (0, 0, 1.5), (-1, 0, 0) };6 7V3 total = (0, 0, 0);8foreach (V3 run in runs) total = Add(total, run);9 10V3 start = (10, 4, 2.5);11Console.WriteLine($"Total displacement: {total}");12Console.WriteLine($"Pipe ends at: {Add(start, total)}");

Total displacement: (2, 2, 1.5)
Pipe ends at: (12, 6, 4)

Subtracting: where is it relative to me?

Subtraction answers “where is B as seen from A?”. Setting-out drawings are full of these: a column’s offset from a grid intersection, a sleeve’s position relative to a slab corner, a fixture’s position relative to a room origin.

1using V2 = (double X, double Y);2 3V2 gridC3 = (12, 6);4V2 column = (12.5, 7.25);5 6V2 offset = (column.X - gridC3.X, column.Y - gridC3.Y);7Console.WriteLine($"Column is {offset} from grid C/3");

Column is (0.5, 1.25) from grid C/3

Scaling: repeats and arrays

Scaling a vector by a whole number repeats a step. An “array” command in a BIM tool is exactly this: the i-th copy sits at start + i × spacing. A two-way grid uses two spacing vectors, one per direction.

1using V2 = (double X, double Y);2 3static V2 Add(V2 a, V2 b) => (a.X + b.X, a.Y + b.Y);4static V2 Scale(V2 v, double k) => (v.X * k, v.Y * k);5 6V2 origin = (0, 0);7V2 along = (7.5, 0);      // bay spacing in X8V2 across = (0, 6);       // bay spacing in Y9 10for (int j = 0; j < 2; j++)11{12    var row = new List<string>();13    for (int i = 0; i < 3; i++)14        row.Add(Add(origin, Add(Scale(along, i), Scale(across, j))).ToString());15    Console.WriteLine(string.Join("  ", row));16}

(0, 0)  (7.5, 0)  (15, 0)
(0, 6)  (7.5, 6)  (15, 6)

The rules you can rely on

• Order does not matter for adding: a + b = b + a, and (a + b) + c = a + (b + c).
• Subtracting is adding the opposite: a − b = a + (−1)b. But a − b = −(b − a), so order matters here.
• Scaling spreads over adding: k(a + b) = ka + kb. Scaling everything about the origin keeps every shape the same, just bigger or smaller.
• The zero vector (0, 0, 0) is “no step”. It is the only vector with no direction, and it will need special care when you normalise.

1using V3 = (double X, double Y, double Z);2 3static V3 Add(V3 a, V3 b) => (a.X + b.X, a.Y + b.Y, a.Z + b.Z);4static V3 Scale(V3 v, double k) => (v.X * k, v.Y * k, v.Z * k);5 6V3 bottom = (2, 5, 0);7V3 step = (0.28, 0, 0.175);8V3 top = Add(bottom, Scale(step, 12));9 10Console.WriteLine($"Top: ({top.X:F3}, {top.Y:F3}, {top.Z:F3})");11Console.WriteLine($"Going: {12 * step.X:F2} m, rise: {12 * step.Z:F2} m");

Top: (5.360, 5.000, 2.100)
Going: 3.36 m, rise: 2.10 m