4.2.8.1 - Vectors, translation and dot products (Python)

List, array and function representations

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

For example, $[2.0,3.14159,-1.0,2.718281828]$ is a four-component vector over $\mathbb{R}$, written as an element of $\mathbb{R}^{4}$. Its dimension is four, regardless of the component values. A Python list [2.0, 3.14159, -1.0, 2.718281828] is a practical representation; a one-dimensional numeric array also stores one component at each index. Python's list does not enforce a common field, so the program must maintain numerical components consistently.

A vector can instead be viewed as a function from its index set to the field:

Here $0\mapsto2.0$, $1\mapsto3.14159$, $2\mapsto-1.0$ and $3\mapsto2.718281828$; $\mapsto$ means “maps to”. A dictionary {0: 2.0, 1: 3.14159, 2: -1.0, 3: 2.718281828} represents that mapping. The index set is the domain and $\mathbb{R}$ is the codomain; it does not mean this one vector contains every real value. All required indices must exist, and arithmetic must match corresponding indices. Dictionary insertion order is not the mathematical definition of a component.

Geometric vectors

A two-component vector can describe a point's position relative to the origin or a displacement from one point to another.

The vector $u=[2,3]$ can be drawn as an arrow from $(0,0)$ to $(2,3)$: two units in the x direction and three in y. Its length is $\lVert u\rVert=\sqrt{2^2+3^2}=\sqrt{13}$. A displacement arrow with the same components can begin elsewhere: from $(1,1)$, it ends at $(3,4)$. Its components describe change, not its starting location.

[DIAGRAM: asset_slug: vector_coordinates; description: Exact coordinate arrows for u equals 2,3 and v equals 4,1, their sum 6,4 and a convex combination at 3.5,1.5]

The blue arrow shows $u$; the green arrow shows $v=[4,1]$. Arrow direction, endpoint and axis scale communicate the components. Negative components move in the negative axis direction. A zero vector has zero length and no defined direction. Higher-dimensional vectors still have components and algebraic operations even when they cannot be pictured on ordinary two-dimensional axes. Sketching an arrow can support your reasoning; assessment here uses supported numerical/source responses rather than scored freehand drawing.

Addition, scaling and convex combinations

Vector arithmetic combines matching components, so the vectors must have the same dimension.

For $u=[2,3]$ and $v=[4,1]$, $u+v=[6,4]$. Adding a displacement translates a position: every point receives the same component changes. Scalar-vector multiplication scales components. $2u=[4,6]$ doubles the length; a negative scalar also reverses direction, while zero produces the zero vector. A positive scalar below one shortens the arrow.

A convex combination has the form $\alpha u+\beta v$, with $\alpha,\beta\geq0$ and $\alpha+\beta=1$. Thus $0.25u+0.75v=[3.5,1.5]$, the purple point on the diagram's segment between the endpoints of u and v. Coefficients 0 and 1 include the endpoints. Coefficients that sum to one but include a negative value give an affine combination outside that segment, not a convex combination. Non-negative coefficients whose sum is not one also fail the definition.