Mathematical Methods
What the exam tests
This domain supports physics numericals: vectors, dot and cross products, differentiation, integration, trigonometry, logarithms, graph slope and area. Questions may ask for direction, angle between vectors, instantaneous rate or area under a graph.
Core rules
- Vector addition uses components. Dot product A·B = AB cosθ is a scalar; cross product |A×B| = AB sinθ is a vector perpendicular to both by right-hand rule.
- Unit vectors i, j, k are mutually perpendicular. A = Ax i + Ay j + Az k.
- Differentiation gives rate of change: d(x^n)/dx = nx^(n−1), d(sin x)/dx = cos x, d(cos x)/dx = −sin x.
- Integration gives accumulated change: ∫x^n dx = x^(n+1)/(n+1) + C. Definite integrals give area.
- On a graph, slope is dy/dx and area under the curve is ∫y dx.
- Useful identities: sin^2θ + cos^2θ = 1; sin 2θ = 2 sinθ cosθ.
Worked example
Given A = 3i + 4j and B = 2i − j, find the dot product and angle. A·B = (3)(2) + (4)(−1) = 6 − 4 = 2. |A| = 5 and |B| = √5 ≈ 2.236. Thus cosθ = 2/(5×2.236) = 0.179, giving θ ≈ 79.7°. The cross product is (3×−1 − 4×2)k = −11k, a vector along the negative z-axis.
Traps
Dot product is scalar; cross product is vector. The angle in A·B is the angle between the vectors placed tail to tail. Do not drop the negative sign in cross products. Differentiation of cos gives minus sin. Area under a velocity-time graph is displacement, while slope is acceleration. Keep units attached to physical quantities.