Geometry / Algebra (Higher) · Higher

Vectors

Column vectors, adding and subtracting vectors, scalar multiples, magnitude, and simple geometric proofs with vectors.

Vectors have both magnitude and direction. In GCSE they are usually written as column vectors. Higher tier expects you to calculate with vectors and use them in geometry.

Column vectors

A vector [a / b] means: • a units right (or left if negative) • b units up (or down if negative) Equal vectors have the same magnitude and direction (same column vector). Opposite vectors: [a / b] and [−a / −b]

Worked example

Write the vector from point A(1, 2) to point B(5, −1) as a column vector.

Solution

B − A = (5 − 1, −1 − 2) = **[4 / −3]**

Adding, subtracting and scalar multiples

[a / b] + [c / d] = [a+c / b+d] [a / b] − [c / d] = [a−c / b−d] k[a / b] = [ka / kb] Resultant vector: the single vector that replaces a combination of vectors.

Worked example

If a = [3 / 1] and b = [−1 / 4], find 2a + b.

Solution

2a = [6 / 2] 2a + b = [6−1 / 2+4] = **[5 / 6]**

Magnitude of a vector

Magnitude (length) of [a / b] = √(a² + b²) This comes from Pythagoras.

Worked example

Find the magnitude of the vector [6 / −8].

Solution

√(6² + (−8)²) = √(36 + 64) = √100 = **10**

Practice questions

Try these without looking at the answers first.

Q1.Vector from (2, 3) to (7, 1) as a column vector?

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[5 / −2]

Q2.a = [2 / −3], b = [4 / 1]. Find a + 2b.

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[10 / −1]

Q3.Magnitude of [3 / 4]?

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