A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Vectors in Pure Mathematics and Contexts — mark scheme explained
The short answer
Vectors are a fundamental concept in mathematics, used to represent quantities that have both magnitude and direction. In AQA A-Level Mathematics, vectors are not only essential for solving problems in pure mathematics but also play a crucial role in applied contexts such as forces and kinematics.
The question
A particle moves with a constant velocity of ¿ v = (2, -1, 3) m/s. If its initial position is ¿r 0 = (1, 2, 4) m, find its position after 5 seconds. [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
The position vector ¿r at time t is given by ¿r = ¿r 0 + ¿ v t.
- S2
Substitute the given values: ¿r = (1, 2, 4) + (2, -1, 3) × 5.
- S3
Calculate the product: (2, -1, 3) × 5 = (10, -5, 15).
- S4
Add the vectors: ¿r = (1 + 10, 2 - 5, 4 + 15) = (11, -3, 19).
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
The position vector ¿r at time t is given by ¿r = ¿r 0 + ¿ v t.
- S2
Substitute the given values: ¿r = (1, 2, 4) + (2, -1, 3) × 5.
- S3
Calculate the product: (2, -1, 3) × 5 = (10, -5, 15).
- S4
Add the vectors: ¿r = (1 + 10, 2 - 5, 4 + 15) = (11, -3, 19).
Final answer: ¿r = (11, -3, 19) m
Common mistakes
- Forgetting to reverse the direction of a vector when subtracting it. — Always remember that ¿a - ¿ b is equivalent to ¿a + (-¿ b). Visualize the vectors geometrically to avoid confusion.
- Incorrectly calculating the dot product by adding magnitudes instead of components. — Always use the formula ¿a ⋅ ¿ b = a 1 b 1 + a 2 b 2 + a 3 b 3 . Double-check your calculations.
- Forgetting to include the initial position when calculating the position vector in kinematics. — Always use the formula ¿r = ¿r 0 + ¿ v t. Ensure you include the initial position ¿r 0 in your calculations.
- Incorrectly interpreting the direction of a unit vector. — Remember that a unit vector has a magnitude of 1 and indicates direction. To find the unit vector in the direction of ¿ v, use ¿e v = ¿ v / |¿ v|.
- Forgetting to check units and dimensions in applied contexts. — Always ensure that all vectors and scalars have consistent units. For example, if velocity is given in m/s, time should be in seconds, and position in meters.
- Incorrectly calculating the magnitude of a vector by adding components instead of squaring them. — Always use the formula |¿a| = √(a 1 2 + a 2 2 + a 3 2 ). Double-check your calculations to ensure you are squaring the components.
Where the marks go
- Full worked solution (all marking points)4 marks