4.2.8.1 - Vectors, translation and dot products (VB.NET)

List, array and function representations

A vector is an ordered collection of components drawn from one field, such as the real numbers.

The list $[2.0,3.14159,-1.0,2.718281828]$ represents a four-vector over $\mathbb{R}$, written as an element of $\mathbb{R}^{4}$. Order matters: swapping the first two entries changes the vector. All four entries come from the same field; mixing a String label with numeric components does not form a real vector.

The same vector has a function interpretation: $f:S\to\mathbb{R}$, where $S=\{0,1,2,3\}$ is the domain and $\mathbb{R}$ is the codomain. Its mappings are $0\mapsto2.0$, $1\mapsto3.14159$, $2\mapsto-1.0$ and $3\mapsto2.718281828$. The symbol $\mapsto$ means “maps to”; indices choose components rather than representing separate geometric axes in code.

VB.NET can hold the same ordered components in a list or one-dimensional array, or use a dictionary to make the function's index-to-value mapping explicit. These declarations belong inside a procedure with System.Collections.Generic imported.

Dim components As New List(Of Double) From {2.0, 3.14159, -1.0, 2.718281828}
Dim values() As Double = {2.0, 3.14159, -1.0, 2.718281828}
Dim mapping As New Dictionary(Of Integer, Double) From {
    {0, 2.0}, {1, 3.14159}, {2, -1.0}, {3, 2.718281828}}
Dim example(3) As Single

example(3) allocates four entries, initially zero; it does not initialise the example's values. Single and Double store finite approximations to real numbers. The mathematical vector's field remains $\mathbb{R}$, not the finite set of representable floating-point values. A complete dictionary representation needs exactly the intended index mappings; dictionary enumeration order is not the component order.

Geometric vectors

A two-component vector can describe a point or the displacement represented by an arrow.

For $u=[2,3]$, draw the arrow from origin $(0,0)$ to $(2,3)$: two units right and three units up. The same displacement can start elsewhere. From $(4,1)$, it ends at $(6,4)$. The starting location changes, but the arrow's components, direction and magnitude remain the same.

A point vector records position relative to an origin; subtracting point vectors gives a displacement between points. The length of $u$ is $\lVert u\rVert=\sqrt{2^2+3^2}=\sqrt{13}$. For $[x,y]$, the length is $\sqrt{x^2+y^2}$. A negative component points left or down; the magnitude cannot be negative. A zero vector has no direction and cannot define an angle through the usual division-by-length formula.

Addition, scaling and convex combinations

Vector arithmetic works component by component and has a geometric meaning.

With $u=[2,3]$ and $v=[4,1]$, $u+v=[6,4]$. Adding displacement $v$ translates a point at $u$ to the new position $[6,4]$, or can be visualised by placing arrows head to tail. Multiplying by 2 scales $u$ to $[4,6]$. A positive scalar preserves direction, a negative scalar reverses it, and zero gives the zero vector. Addition requires matching dimensions.