Dot product, norms & angles
The dot product u·v = Σuᵢvᵢ also equals |u||v|cos θ, which is what makes it geometric: it is zero exactly when the vectors are perpendicular, positive for acute angles and negative for obtuse. The norm |v| = √(v·v) generalises length, and the Cauchy–Schwarz inequality |u·v| ≤ |u||v| is what guarantees the cosine formula makes sense.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Compute the dot product componentwiseMultiply matching components and add. The result is a scalar, not a vector.
- Compute the norms|v| = √(v₁² + v₂² + …). Keep them in surd form for exact answers.
- Solve for the anglecos θ = (u·v)/(|u||v|), then θ = arccos of that. The value always lies in [−1, 1] by Cauchy–Schwarz.
- Read off perpendicularityu·v = 0 means orthogonal. This is the quickest test and needs no norms at all.
Worked example
Find the angle between u = (1, 2, 2) and v = (2, 0, 1).
- u·v = 1·2 + 2·0 + 2·1 = 4.
- |u| = √(1 + 4 + 4) = 3.
- |v| = √(4 + 0 + 1) = √5.
- cos θ = 4/(3√5) ≈ 0.5963.
Answer. θ = arccos(4/(3√5)) ≈ 53.4°.
Where marks get dropped
These are the specific errors that cost credit on dot product, norms & angles questions — QED's rubric penalises each of them separately.
- Treating the dot product as a vector. It is a scalar — the cross product is the one that returns a vector, and only in ℝ³.
- Forgetting the square root in the norm, which makes every angle wrong.
- Reporting a cosine outside [−1,1], which always indicates an arithmetic error rather than an exotic geometry.
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Dot product, norms & angles — frequently asked questions
What does a negative dot product mean?
The angle is obtuse — the vectors point in broadly opposite directions. Zero means perpendicular, positive means acute.
What is Cauchy–Schwarz good for?
It guarantees |u·v|/(|u||v|) lies in [−1,1] so the arccos is defined, and it is the source of the triangle inequality |u+v| ≤ |u| + |v|.
How does this generalise?
Any inner product on a vector space — including ∫f g for functions — gives norms, angles and orthogonality. Fourier series are exactly an orthogonal decomposition in such a space.
The rest of Linear Algebra
Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Matrix arithmetic & inverses
- 2Gaussian elimination
- 3Determinants
- 4Vector spaces, span & linear independence
- 5Basis & dimension
- 6Eigenvalues & eigenvectors
- 7Linear maps & their matrices
- 8Rank, nullity & the rank–nullity theorem
- 9Column space, null space & solution sets
- 10Orthogonality, projections & Gram–Schmidt
- 11Diagonalisation & powers of a matrix
- 12Dot product, norms & angles
- 13Change of basis & similar matrices
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