Connect the ideas.
Position
Each coordinate uses the same parameter.
Motion
Speed is the magnitude .
Gradient
Use the quotient only when .
Arc length
Integrate speed over the stated interval.
- A Cartesian equation describes a locus; retain the parameter interval, starting point, orientation and any repeated tracing.
- A collision needs equal position vectors at one common time. Intersecting paths can be visited at different times.
- Differentiate or integrate each component; use every initial condition to determine the integration constants.
- Velocity gives direction and rate of motion. Speed is its nonnegative magnitude; a zero coordinate rate alone does not imply rest.
- At a nonzero derivative vector, the tangent follows that vector. Check zero components before dividing; a zero vector needs a separate limiting argument.
- Bézier endpoint controls lie on the path; interior controls set its shape. Matching tangent directions alone does not guarantee equal endpoint derivatives.
- Acceleration changes velocity. Constant speed can have nonzero acceleration when the direction changes.
- Arc length integrates the magnitude of the parameter derivative. Travelled distance counts retracing; displacement depends only on the endpoints.
- For proofs, state the interval and any nonzero assumptions, justify signs before removing absolute values, and establish claims throughout the stated domain.
Put it together.
Choose a paper and show your working. Open its marking guide when you are ready to check.
Vector calculus — Test 1
Foundations, interpretation and increasingly connected motion problems.
10 questions · 58 marks · 90 minutes · Scientific calculatorVector calculus — Test 2
Rates, tangent directions, collisions and mathematical justification.
10 questions · 62 marks · 90 minutes · Scientific calculatorVector calculus — Test 3
Polynomial curves and Bézier design, followed by connected modelling tasks.
10 questions · 63 marks · 90 minutes · Scientific calculatorVector calculus — Test 4
Spatial motion, circular motion and the geometry of acceleration.
10 questions · 64 marks · 90 minutes · Scientific calculatorVector calculus — Test 5
Mixed applications, nonuniform parameterisations and optimisation.
10 questions · 66 marks · 90 minutes · Scientific calculatorVector calculus — Test 6
Synthesis, proofs and unfamiliar but hand-solvable applications.
10 questions · 67 marks · 90 minutes · Scientific calculator
Curriculum reference & topic scope
SACE Board of South Australia
Stage 2 Specialist Mathematics school Vector calculus: polynomial and trigonometric parameterisations, related coordinate rates, Bézier paths, componentwise differentiation and integration, motion, tangent directions, circular motion and arc length. Reasoning and proofs use these school topics; vector-field calculus is outside this chapter.
Official curriculum source ↗