Higher Maths · Expressions and Functions
Vectors | Higher Maths
Vectors in Higher Maths extend National 5 work into 3D and introduce the scalar (dot) product. Expect a 6–8 mark vector question on Paper 2, often involving showing that three points are collinear or finding the angle between two vectors.
Vector essentials
Magnitude: |v| = √(a2 + b2 + c2) for v = (a, b, c)
Unit vector: v̂ = v / |v|
Scalar (dot) product: a · b = a1b1 + a2b2 + a3b3
Angle between vectors: cos θ = (a · b) / (|a| |b|)
Perpendicular vectors: a · b = 0
Worked example
Worked example — Angle between two vectors
Problem: Find the angle between a = (2, −1, 2) and b = (1, 2, 2).
- Compute the scalar product.a · b = 2(1) + (−1)(2) + 2(2) = 2 − 2 + 4 = 4
- Compute magnitudes.|a| = √(4 + 1 + 4) = 3 ; |b| = √(1 + 4 + 4) = 3
- Apply the formula.cos θ = 4 / (3 × 3) = 4/9
- Solve.θ = cos−1(4/9) ≈ 63.6°
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Find |(3, −4, 12)|.
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√(9 + 16 + 144) = 13 - Show that (1, 2, 3) and (2, −1, 0) are perpendicular.
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Dot product = 2 − 2 + 0 = 0. Perpendicular. - Find the unit vector in the direction of (4, 0, −3).
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|v| = 5. Unit vector = (4/5, 0, −3/5) - Show that A(1, 2, 3), B(2, 4, 5) and C(4, 8, 9) are collinear.
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AB = (1, 2, 2), BC = (2, 4, 4) = 2AB. Same direction with shared point B ⇒ collinear. - If a = (1, −2, 2) and b = (3, 0, −1), find a · b.
Show answer
3 + 0 − 2 = 1
Common mistakes
Common mistakes & how to avoid them
- Forgetting to include all three components in the dot product when working in 3D.
- Confusing the dot product (scalar) with the cross product (vector — not on Higher).
- Failing to include "shared point" reasoning when proving collinearity.
Frequently asked questions
Are vector formulas on the SQA formula sheet?
The dot product formula is provided. Magnitude and the angle formula are not, so memorise them.
What is the difference between a position vector and a displacement vector?
A position vector goes from the origin to a point. A displacement vector goes from one point to another. AB = b − a.
Can vectors be negative?
Components can be negative; magnitude is always non-negative.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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