A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Vector Addition and Scalar Multiplication — mark scheme explained
The short answer
Vectors are mathematical objects that have both magnitude (size) and direction. In A-Level Mathematics, you will learn how to add vectors diagrammatically and perform algebraic operations such as vector addition and scalar multiplication. Understanding these operations is crucial for solving problems in physics, engineering, and other fields.
The question
Given ·α = (2, 5) and ·β = (-1, 3), find ·α + ·β. [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
Step 1: Identify the components of each vector. - ·α = (2, 5) - ·β = (-1, 3)
- S2
Step 2: Add the corresponding components. - x-component: 2 + (-1) = 1 - y-component: 5 + 3 = 8
- S3
Step 3: Write the resultant vector. - ·α + ·β = (1, 8)
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Step 1: Identify the components of each vector. - ·α = (2, 5) - ·β = (-1, 3)
- S2
Step 2: Add the corresponding components. - x-component: 2 + (-1) = 1 - y-component: 5 + 3 = 8
- S3
Step 3: Write the resultant vector. - ·α + ·β = (1, 8)
Final answer: (1, 8)
Common mistakes
- Adding the wrong components when performing vector addition. — Always double-check that you are adding the corresponding x-components and y-components separately.
- Forgetting to reverse the direction when multiplying by a negative scalar. — Remember that multiplying by a negative scalar reverses the direction of the vector. Always check the sign of the scalar.
- Incorrectly drawing the parallelogram method for vector addition. — Ensure that you draw lines parallel to each vector and complete the parallelogram accurately. The resultant vector should be the diagonal from the common tail to the opposite corner.
- Forgetting to multiply all components by the scalar in scalar multiplication. — Always multiply each component of the vector by the scalar. Double-check your work to ensure all components are correctly multiplied.
- Incorrectly identifying the tail and head of vectors in the triangle method. — Place the tail of the second vector at the head of the first vector. The resultant vector should start from the tail of the first vector and end at the head of the second vector.
- Forgetting to write the final answer in component form. — Always write your final answer in component form (x, y) unless otherwise specified.
- Incorrectly calculating the magnitude of a vector after scalar multiplication. — Double-check your calculations for squaring and adding the components. Use a calculator if necessary to avoid simple arithmetic mistakes.
Where the marks go
- Full worked solution (all marking points)3 marks