A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Position Vectors and Distance Between Points — mark scheme explained
The short answer
In AQA A-Level Mathematics, understanding position vectors and calculating the distance between two points represented by these vectors is a crucial skill. This section will cover what position vectors are, how to represent them, and the method for finding the distance between two points using vector notation. What Are Position Vectors?
The question
Point A has coordinates (1, 3) and point B has coordinates (4, 7). Calculate the distance between points A and B. [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
Find the position vectors OA and OB :
- S2
OA = 1 i + 3 j
- S3
OB = 4 i + 7 j
- S4
Calculate the vector AB :
- S5
AB = (4 - 1) i + (7 - 3) j = 3 i + 4 j
- S6
Find the magnitude of AB :
- S7
| AB | = √(3 2 + 4 2 ) = √(9 + 16) = √25 = 5 units
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Find the position vectors OA and OB :
- S2
OA = 1 i + 3 j
- S3
OB = 4 i + 7 j
- S4
Calculate the vector AB :
- S5
AB = (4 - 1) i + (7 - 3) j = 3 i + 4 j
- S6
Find the magnitude of AB :
- S7
| AB | = √(3 2 + 4 2 ) = √(9 + 16) = √25 = 5 units
Final answer: 5 units
Common mistakes
- Forgetting to subtract the position vectors correctly when finding AB — Always ensure that you subtract the coordinates of point A from those of point B: AB = OB - OA .
- Using the wrong formula for the magnitude of a vector — Use the correct formula for the magnitude of a vector: | v | = √( a 2 + b 2 + c 2 ) in three dimensions.
- Forgetting to take the square root when calculating the magnitude — Always remember to take the square root after squaring and summing the components: | v | = √( a 2 + b 2 + c 2 ).
- Confusing the position vector with the coordinates of a point — Always represent the position vector using unit vectors: OP = x i + y j + z k .
- Using the wrong components when calculating the magnitude — Double-check that you are using the correct components for each term in the formula: | v | = √( a 2 + b 2 + c 2 ).
- Forgetting to include all components in three-dimensional space — Ensure that you include all components (x, y, and z) when calculating the magnitude of a vector in three-dimensional space.
Where the marks go
- Full worked solution (all marking points)4 marks