4.2.8.1 - Vectors, translation and dot products (Java)

List, array and function representations

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

The vector $[2.0,3.14159,-1.0,2.718281828]$ is a four-vector over $\mathbb{R}$, written as an element of $\mathbb{R}^{4}$. Its dimension is four, regardless of the values or the amount of memory used. A Java double approximates real values with finite precision; it does not implement every real number exactly.

The same vector can be interpreted as a function $f:S\to\mathbb{R}$, where $S=\{0,1,2,3\}$. Its mappings are $0\mapsto2.0$, $1\mapsto3.14159$, $2\mapsto-1.0$ and $3\mapsto2.718281828$. The symbol $\mapsto$ means maps to. The input is a component index and the output is that component's value; this is not a function taking an arbitrary real input.

// import java.util.*;
double[] array = {2.0, 3.14159, -1.0, 2.718281828};
List<Double> list = Arrays.asList(2.0, 3.14159, -1.0, 2.718281828);
Map<Integer, Double> function = new HashMap<>();
for (int i = 0; i < array.length; i++) function.put(i, array[i]);
double third = function.get(2); // -1.0

The array and ordered list preserve component order directly. The dictionary preserves the mapping through keys even if iteration order changes. To represent this whole four-vector, every index in the domain must have its specified component; an omitted key is not automatically zero. The representation is Java-specific, while the vector and function meanings are mathematical.

Geometric vectors

A two-vector can represent a point or a displacement in a coordinate plane.

The vector $\mathbf{u}=[2,3]$ is drawn with tail at the origin and head at $(2,3)$: two units across and three up. Moving the arrow's tail does not change the displacement if its direction and length remain the same. A point uses a position relative to an origin; a displacement describes a change in position.

[DIAGRAM: asset_slug: vector_coordinates; description: Exact coordinate plot of u equals (2,3), v equals (4,1), their sum (6,4), and convex point (3.5,1.5).]

The magnitude is $\lVert\mathbf{u}\rVert=\sqrt{2^2+3^2}=\sqrt{13}$. Negative components point in the corresponding negative direction. A zero vector has zero magnitude and no direction, so an angle involving it cannot be found by the nonzero-vector formula.

Addition, scaling and convex combinations

Vector operations combine corresponding components and require matching dimensions.

For $\mathbf{u}=[2,3]$ and $\mathbf{v}=[4,1]$, their sum is $[6,4]$. Adding $\mathbf{v}$ translates the point $(2,3)$ by four across and one up. Multiplying $\mathbf{u}$ by 2 gives $[4,6]$, scaling its magnitude by 2; a negative scalar also reverses direction and zero gives the zero vector.

A convex combination is $\alpha\mathbf{u}+\beta\mathbf{v}$, where $\alpha,\beta\geq0$ and $\alpha+\beta=1$. It gives a point on the segment between the two vector positions, including the endpoints. For $\alpha=0.25$ and $\beta=0.75$, the result is $[0.5+3,0.75+0.75]=[3.5,1.5]$. Coefficients summing to one alone are insufficient: a negative coefficient is not convex.

static double[] add(double[] u, double[] v) {
    if (u.length != v.length) throw new IllegalArgumentException("Dimension");
    double[] result = new double[u.length];
    for (int i = 0; i < u.length; i++) result[i] = u[i] + v[i];
    return result;
}