📘 GCSE Maths · Notes & worksheet
Vectors.
Notes
Basics
- A vector has magnitude and direction. In bold or underlined notation: a, or AB with an arrow.
- Column form: (3, −2) means 3 right and 2 down.
- Magnitude of (x, y) is √(x² + y²).
- Add vectors by adding components. Multiply by a scalar by multiplying each component.
- Reverse direction = negative: BA = −AB.
Parallel vectors
Two vectors are parallel if one is a scalar multiple of the other. For example (6, −9) = −1.5 × (−4, 6).
Collinear points
Points A, B, C are collinear if AB is parallel to AC, because they share the point A.
Geometric proof method
- Write every unknown vector as a journey along known vectors: for example AB = AO + OB = −a + b.
- Use ratios: if P divides AB in the ratio 2:3, then AP = (2/5)AB.
- To prove lines are parallel, show one vector is a multiple of the other.
- To prove three points are collinear, show two vectors are parallel and share a point.
Worked example
OAB is a triangle with OA = a and OB = b. P lies on AB with AP:PB = 2:3.
- AB = b − a
- AP = (2/5)(b − a)
- OP = OA + AP = a + (2/5)(b − a) = (3/5)a + (2/5)b
Common mistakes
- Getting directions backwards: AB means "from A to B".
- Forgetting to simplify the final expression in terms of a and b.
- Claiming collinear without saying the vectors share a point.
Exam tips
- Always draw the journey on the diagram.
- "Show that" proofs need a clear conclusion sentence, such as "so MN is parallel to AB and half its length".
Worksheet
Prints the questions only, without answers. Choose “Save as PDF” in the print window to download.
Diagram placeholderThe diagrams for Q3–Q7 are described in words in the questions. The drawn diagrams are being added to this page.
Total: 19 marks.
- a = (3, −2) and b = (−1, 4).
(a) Find 2a − 3b. (2)
(b) Find the exact magnitude of a + b. (1) - Are the vectors (6, −9) and (−4, 6) parallel? Give a reason. (2)
- OABC is a parallelogram with OA = a and OC = c. M is the midpoint of AB. Find OM in terms of a and c. (2)
- OAB is a triangle with OA = a and OB = b. P is on AB such that AP:PB = 2:3. Find OP in terms of a and b. (3)
- In triangle OAB, OA = a and OB = b. M is the midpoint of OA and N is the midpoint of OB. Prove that MN is parallel to AB and half its length. (3)
- Points A, B and C have position vectors (1, 2), (4, 8) and (−1, −2). Show that A, B and C are collinear. (3)
- In triangle OAB, OA = a and OB = b. M is the midpoint of AB. G lies on OM with OG:GM = 2:1. Find OG in terms of a and b. (3)
Show mark scheme and answers (tutor copy)
Try the worksheet first, then check your work here.
- (a) 2(3, −2) − 3(−1, 4) = (6, −4) − (−3, 12) = (9, −16). (M1, A1)
(b) a + b = (2, 2), so the magnitude is √(4 + 4) = 2√2. (B1) - (6, −9) = −1.5 × (−4, 6), so yes, they are parallel because one is a scalar multiple of the other. (M1, A1)
- AB = OC = c, since OABC is a parallelogram. OM = OA + ½AB = a + ½c. (M1, A1)
- AB = b − a (B1). AP = (2/5)(b − a) (M1). OP = a + (2/5)(b − a) = (3/5)a + (2/5)b. (A1)
- OM = ½a and ON = ½b (B1). MN = MO + ON = −½a + ½b = ½(b − a) (M1). AB = b − a, so MN = ½AB. MN is a multiple of AB, so they are parallel and MN is half the length. (A1)
- AB = (3, 6) and AC = (−2, −4) (M1). (3, 6) = −1.5 × (−2, −4) (M1), so AB is parallel to AC. They share the point A, so A, B and C are collinear. (A1)
- OM = OA + ½AB = a + ½(b − a) = ½(a + b) (M1, A1). OG = (2/3)OM = (1/3)(a + b). (A1)