Step-by-step worked examples and graded practice questions on vectors — notation, addition, subtraction, scalar multiplication, magnitude, and vector geometry proofs.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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A vector represents a movement — it has both a size (magnitude) and a direction. A vector is written as a pair of numbers showing the horizontal and vertical movement, e.g. a = (42) means "move 4 right, 2 up". A vector from point A to point B can also be written as AB.
Worked Example 1 — Vector notation
Write down the vector that represents the movement from A(1,1) to B(5,3).
1
Find the horizontal change: 5 − 1 = 4
2
Find the vertical change: 3 − 1 = 2
AnswerAB = (42)
The vector a runs from A to B — 4 units right and 2 units up.
Addition, subtraction and scalar multiplication
To add or subtract vectors, add or subtract the corresponding components. To multiply a vector by a scalar (an ordinary number), multiply every component by that number — this scales the vector's length and, if the scalar is negative, reverses its direction.
Worked Example 2 — Adding vectors
a = (31) and b = (−14). Find a + b.
1
Add the horizontal components: 3 + (−1) = 2
2
Add the vertical components: 1 + 4 = 5
Answera + b = (25)
The triangle law of addition: travelling along a then b (teal, then black) ends at the same point as travelling directly along a+b (amber, dashed).
Worked Example 3 — Scalar multiplication
a = (21). Find 3a and −a, and describe each geometrically.
1
3a = (3×23×1) = (63) — three times as long, same direction as a
2
−a = (−2−1) — the same length as a, but pointing in the opposite direction
Answer3a = (63); −a = (−2−1)
3a points the same way as a but is three times as long; −a is the same length as a but points the opposite way.
Magnitude of a vector
The magnitude of a vector is its length, written |a|. Since a vector's horizontal and vertical components form a right-angled triangle with the vector itself as the hypotenuse, the magnitude is found using Pythagoras' theorem: |a| = √(x² + y²)
Worked Example 4 — Magnitude
Find the magnitude of the vector a = (512).
1
|a| = √(5² + 12²) = √(25 + 144) = √169
Answer|a| = 13
The vector (512) is the hypotenuse of a right-angled triangle with legs 5 and 12 — its magnitude follows from Pythagoras' theorem.
Vector geometry
Vectors can prove geometric facts without any coordinates or measuring. The key idea: if AB = k × CD for some number k, then AB is parallel to CD (and if they share a point, they're collinear). The midpoint of a line from position vector a to position vector b has position vector ½(a + b).
Worked Example 5 — Midpoint
O is the origin. A has position vector a = (62) and B has position vector b = (26). M is the midpoint of AB. Find the position vector of M.
1
The midpoint of AB has position vector OM = ½(a + b)
2
OM = ½((62) + (26)) = ½(88)
AnswerOM = (44)
M is the midpoint of AB, so OM = ½(a+b).
Practice questions
Work through each question before checking the answers. Where a question shows points or vectors on a diagram, the diagram is required to answer it.
Foundation (Grade 3–5)
Q1Write down the column vector for the arrow shown, from P to Q.Foundation
Diagram for Question 1 (not to scale).
Q2a = (4−2). Find 2a.Foundation
Q3a = (35) and b = (1−2). Find a + b.Foundation
Q4Find the magnitude of the vector (68).Foundation
Q5a = (53). Find −a.Foundation
Higher (Grade 5–7)
Q6a = (7−1) and b = (−34). Find a − b.Higher
Q7Find the magnitude of the vector (912).Higher
Q8O is the origin. A has position vector a = (26) and B has position vector b = (82). M is the midpoint of AB. Find the position vector of M.Higher
Diagram for Question 8 (not to scale).
Q9a = (25) and b = (4−1). Find 3a − 2b.Higher
Q10The diagram shows vectors AB and CD. Explain why AB is parallel to CD.Higher
Diagram for Question 10 (not to scale).
Higher — Hard (Grade 8–9)
Q11O is the origin. A has position vector a = (14) and B has position vector b = (101). P lies on AB such that AP : PB = 2 : 1. Find the position vector of P.Grade 8–9
Diagram for Question 11 (not to scale). P divides AB in the ratio 2:1.
Q12OABC is a quadrilateral. OA = a and OC = c. Given that OABC is a parallelogram, write down the position vector of B in terms of a and c.Grade 8–9
Q13Find the magnitude of the vector (2√32), giving your answer as an integer.Grade 8–9
Q14O is the origin. A has position vector a = (82) and B has position vector b = (28). P is the midpoint of OA and Q is the midpoint of OB. Prove that PQ is parallel to AB and find the ratio of their lengths.Grade 8–9
Diagram for Question 14 (not to scale).
Q15Vectors a = (3k) and b = (64) are parallel. Find the value of k.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1(33)
Q2(8−4)
Q3(43)
Q410
Q5(−5−3)
Higher (Q6–Q10)
Q6(10−5)
Q715
Q8(54)
Q9(−217)
Q10AB = (63) and CD = (42). Since (63) = 1.5 × (42), AB = 1.5 × CD, so AB is parallel to CD.
Higher — Hard (Q11–Q15)
Q11(72)(OP = a + ⅔(b − a))
Q12OB = a + c
Q134(√(12+4) = √16)
Q14PQ = ½AB, so PQ is parallel to AB and half its length(PQ = q − p = ½b − ½a = ½(b−a) = ½AB)
Q15k = 2(b = 2a, since 6 = 2×3, so 4 = 2k)
Common mistakes
Common Mistake 1
Forgetting vectors have direction, not just size
(34) and (43) are different vectors even though they have the same magnitude — always keep the horizontal and vertical components in the correct order and with the correct sign.
Common Mistake 2
Mixing up the direction of AB and BA
AB and BA are opposite vectors: BA = −AB. Always check which direction the question is asking for.
Common Mistake 3
Assuming vectors that look parallel actually are
To prove two vectors are parallel, show algebraically that one is a scalar multiple of the other (e.g. AB = 2CD) — don't just say they "look" parallel on a diagram.
Common Mistake 4
Losing track of position vectors vs. vectors between points
A position vector (like OA) gives a point's location relative to the origin. A vector like AB describes a movement between two points, found by AB = OB − OA. Confusing the two is a common source of errors in geometry proofs.
Exam tips
💡 Exam Tip 1
Always write vectors in terms of the given letters
In geometry proofs, express every vector you need in terms of the given position vectors (usually a, b, c) before trying to combine or compare them.
💡 Exam Tip 2
Use AB = OB − OA whenever you need a vector between two points
This single rule underpins almost every vector geometry question — memorise it and check it applies before doing anything else.
💡 Exam Tip 3
State the parallel or ratio conclusion explicitly
After showing PQ = k × AB, write the conclusion in words ("so PQ is parallel to AB") — the algebra alone often doesn't earn the final mark without this statement.
💡 Exam Tip 4
Sketch the diagram even if one isn't given
A quick sketch of the points and vectors involved makes it much easier to spot which combination of vectors you need, especially for ratio and midpoint problems.
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