Vectors revision notes

What is a vector?

A vector is an ordered collection of values. All entries must be drawn from the same field, such as the real numbers, ℝ. A 4-vector over ℝ can be written as [2.0, 3.14159, -1.0, 2.718281828]. A 2-vector over ℝ could be [2.0, 3.0].

Different representations

Viewed as a function, the vector can be written as f: S → ℝ, where S = {0, 1, 2, 3} and the co-domain is ℝ. The mappings are:

• 0 ↦ 2.0
• 1 ↦ 3.14159
• 2 ↦ -1.0
• 3 ↦ 2.718281828

The symbol ↦ means “maps to”. A dictionary is useful for this function interpretation. In Python, the same vector is {0: 2.0, 1: 3.14159, 2: -1.0, 3: 2.718281828}. In VB.NET, a 4-vector over ℝ can be represented by the one-dimensional array declaration Dim example(3) As Single.

Geometric interpretation

A vector can be visualised as an arrow. The 2-vector [2.0, 3.0] has its tail at the origin and its head at (2.0, 3.0).

Operations

Vector addition achieves translation. Scalar-vector multiplication achieves scaling. A convex combination of vectors u and v has the form αu + βv, where α ≥ 0, β ≥ 0, and α + β = 1.

Dot product

For u = [u1, …, un] and v = [v1, …, vn], the dot or scalar product is u · v = u1v1 + u2v2 + …… + unvn. For example, [2, 3] · [4, 5] = (2 × 4) + (3 × 5) = 23. The dot product is applied when finding the angle between two vectors.

Common errors

Do not mix values from different fields in one vector. Do not confuse a vector’s list representation with its function interpretation. Remember that vector addition represents translation, whereas scalar-vector multiplication represents scaling. In a dot product, multiply corresponding entries and add the products; do not multiply only the first entries or concatenate the vectors.