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Vectors
Official AQA 7517 section 4.2.8.
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5
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36 min
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AqaA LevelComputer ScienceFundamentals of data structures
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Vectors1 objectives
- Vectors: Be familiar with the concept of a vector and the following notations for specifying a vector: • [2.0, 3.14159, -1.0, 2.718281828] • 4-vector over ℝ written as ℝ4 • function interpretation • 0 ↦ 2.0 • 1 ↦ 3.14159 • 2 ↦ -1.0 • 3 ↦ 2.718281828 • ↦ means maps to That all the entries must be drawn from the same field, eg ℝ. A vector can be represented as a list of numbers, as a function and as a way of representing a geometric point in space. A dictionary is a useful way of representing a vector if a vector is viewed as a function. f: S → ℝ the set S = {0,1,2,3} and the co-domain, ℝ, the set of Reals For example, in Python the 4-vector example could be represented as a dictionary as follows: {0:2.0, 1:3.14159, 2:-1.0, 3:2.718281828} Dictionary representation of a vector. See above. List representation of a vector. For example, in Python, a 2-vector over ℝ would be written as [2.0,3.0]. 1-D array representation of a vector. For example in VB.Net, a 4-vector over ℝ would be written as Dim example(3) As Single. Visualising a vector as an arrow. For example a 2-vector [2.0, 3.0] over ℝ can be represented by an arrow with its tail at the origin and its head at (2.0, 3.0). Vector addition and scalar-vector multiplication. Know that vector addition achieves translation and scalar-vector multiplication achieves scaling. Convex combination of two vectors, u and v. Is an expression of the form αu + βv where α, β ≥ 0 and α + β = 1 Dot or scalar product of two vectors. The dot product of two vectors, u and v, u = [u1, …., un ] and v = [v1, ….., vn ] is u ∙ v = u1v1 + u2v2 + …… + unvn Applications of dot product. Finding the angle between two vectors.
Key terms
Dot productCo-domain
Exam tips
- Check convex-combination conditions explicitly: When identifying or constructing a convex combination, write down both checks: α ≥ 0 and β ≥ 0, then verify that α + β = 1.
Common mistakes
- Accepting mixed entry types in a vector: Check that every entry is drawn from the same field. For a vector over ℝ, every entry must be a real number.
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