0580/42 Oct/Nov 2024, Question 6: Adding and Subtracting Vectors

Part (a) · 2 marks

Subtract the column vector (2, −7) from (3, −5) and give the resulting column vector.

1. 12 × (3, −5) = (6, −10)
2. 2(6, −10) − (2, −7) = (4, −3)

Answercolumn vector (4, −3)

Part (bi) · 1 mark

Given MN = (−6, 4) and M = (2, −5), find the coordinates of point N.

1. 1N = (2 + (−6), −5 + 4) = (−4, −1)

Answer(−4, −1)

Part (bii) · 2 marks

Find the magnitude of the vector MN, where MN = (−6, 4).

1. 1|MN| = √((−6)² + 4²) = √52
2. 2= 7.211… ≈ 7.21

Answer7.21 or 7.211…

Part (cia) · 1 mark

In the trapezium, express vector AB in terms of a and c in simplest form.

1. 1AB = AO + OB = −a + 2c = 2c − a

Answer2c − a

Part (cib) · 1 mark

Express vector CB in terms of a and c in simplest form, using OB = 2AC.

1. 1AC = 1/2 OB = c, so OC = a + c
2. 2CB = OB − OC = 2c − (a + c) = c − a

Answerc − a

Part (cic) · 2 marks

Using AP : PB = 4 : 1, express vector OP in terms of a and c in simplest form.

1. 1AP = 4/5 AB = 4/5(2c − a)
2. 2OP = a + 4/5(2c − a) = 1/5 a + 8/5 c
3. 3OP = (1/5)(a + 8c)

Answer(1/5)(a + 8c)

Part (cid) · 2 marks

Using AQ = 4/5 AC, express vector QP in terms of a and c in simplest form.

1. 1QA = −4/5 AC = −4/5 c
2. 2QP = QA + AP = −4/5 c + 4/5(2c − a)
3. 3QP = 4/5 c − 4/5 a = (4/5)(−a + c)

Answer(4/5)(−a + c)

Part (cii) · 2 marks

From your expressions, state two facts relating lines QP and CB, such as direction and length.

1. 1QP = (4/5)(c − a) and CB = c − a
2. 2So QP is a multiple of CB: QP is parallel to CB and QP = (4/5)CB

AnswerQP is parallel to CB and QP = (4/5)CB

More past-paper questions on adding and subtracting vectors

• 0580/42 Feb/Mar 2024 Q7 · 7 marks
• 0580/12 May/Jun 2024 Q10 · 1 mark
• 0580/21 Oct/Nov 2024 Q5 · 2 marks
• 0580/12 Feb/Mar 2023 Q8 · 2 marks
• 0580/11 May/Jun 2023 Q11 · 2 marks
• 0580/12 May/Jun 2023 Q15 · 3 marks