(Level 6 - 9)

Vectors
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1 𝐾𝐿𝑀 is a scalene triangle. (Level 6)

𝐾
Not drawn
accurately

2𝒂
𝒃

𝑀 𝐿
𝑃

Point P is half way along 𝑀𝐿.

𝑀𝐾 = 𝟐𝒂, 𝐾𝐿 = 𝒃.

1(a) Write the vector 𝑀𝐿 in terms of 𝒂 and 𝒃.

[1 mark]

Answer

1(b) Find 𝑀𝑃 in terms of 𝒂 and 𝒃.

[1 mark]

Answer

1(c) A new point 𝑁 is to the right of 𝐿.

𝑀𝑁 is 4 times the length of 𝑀𝑃.

Write 𝑀𝑁 as a column vector.

[1 mark]

Answer

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2 𝐸𝐹𝐺 is a scalene triangle. (Level 6)

E 3𝒂 + 4𝒃
F

Not drawn
𝒂 − 5𝒃
accurately

𝐸𝐺 = 𝒂 − 5𝒃, 𝐸𝐹 = 3𝒂 + 4𝒃.

𝐺𝐻, not shown, is 3 times the length of and parallel to 𝐺𝐹.

Find 𝐺𝐻 in terms of 𝒂 and 𝒃.

[3 marks]

Answer

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3 𝐴𝐵𝐶𝐷 is a rhombus. Opposite sides are parallel. (Level 7)

a 𝐵 𝐸
𝐴

b Not drawn
accurately

𝐷 𝐹 𝐶

𝐵𝐸 is an extension of 𝐴𝐵, such that 𝐴𝐵 ∶ 𝐵𝐸 = 4: 3

The point F is halfway along 𝐶𝐷

Find 𝐹𝐸 in terms of 𝒂 and 𝒃.

[3 marks]

Answer

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4 KLMN is an irregular quadrilateral. (Level 7)

K

3𝒂 + 𝒃

Not drawn
𝑁
accurately

P
2𝒂 + 2𝒃

L
𝑀 2𝒂

𝑀𝐿 = 𝟐𝒂, 𝑀𝑁 = 2𝒂 + 𝟐𝒃, 𝑁𝐾 = 3𝒂 + 𝒃.
Point P is positioned such that 𝐿𝑃: 𝑃𝐾 = 1: 2

4(a) Find the vector 𝐿𝐾 in terms of 𝒂 and 𝒃.

[2 marks]

Answer

4(b) Show that 𝑀𝑃 = 𝑁𝐾.

[4 marks]

Answer

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5 The diagram below shows a regular hexagon ABCDEF. (Level 7)

X is the point at the center of the hexagon.

𝑋𝐵 = 𝒂

𝐸𝐷 = 𝒃

𝐴
Not drawn
accurately
𝐹 𝐵
𝑎

𝐸 𝐶

𝑏
𝐷

Write down the following vectors in terms of a and b.

5(a) 𝐵𝐶
[1 mark]

5(b) 𝐵𝐸
[1 mark]

5(c) 𝐴𝐸
[1 mark]

5(d) 𝐹𝐵
[1 mark]

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6 𝐴𝐵𝐶 is an isosceles triangle. (Level 7)

E

𝐵 Not drawn
accurately

3a

𝐴 b M 𝐶 𝐷

Point 𝑀 is halfway along the triangle base, 𝐴𝐶.

Point 𝐷 is positioned such that 𝐴𝐶 and 𝐴𝐷 are collinear.

Point 𝐸 is positioned such that 𝐴𝐵 and 𝐴𝐸 are collinear.

𝐴𝐵: 𝐵𝐸 = 3:2

𝑀𝐴 = 𝒃, 𝐴𝐵 = 3𝒂, 𝐴𝐷 = 2𝐴𝐶.

Find 𝐷𝐸 in terms of 𝒂 and 𝒃.

[4 marks]

Answer

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7 On the diagram below: (Level 7)

AE is a straight line

𝐴𝐵 = 𝒂

𝐴𝐶 = 𝒃
𝐴𝐶 = 𝐶𝐸
1
𝐷𝐸 = 𝐶𝐸
4

𝐵
Not drawn
accurately

𝐴
𝑏 𝐶 𝐷 𝐸

Find expressions for the vectors below in terms of a and b.

7(a) 𝐵𝐶
[1 mark]

Answer

7(b) 𝐷𝐵
[4 marks]

Answer

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8 𝐴𝐵𝐶 is a scalene triangle. (Level 8)

B
Not drawn
D accurately

2a

A C
3b

𝐴𝐶 = 3𝒃, 𝐴𝐵 = 2𝒂.
Point D is positioned such that 𝐵𝐷: 𝐷𝐶 = 1: 3
Point E is positioned such that 𝐴𝐸: 𝐸𝐷 = 1: 1

Find 𝐴𝐸 in terms of 𝒂 and 𝒃.

[5 marks]

Answer

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9 On the diagram below: (Level 8)

𝐴𝐵 = 𝐵𝐶 = 𝒂

𝐴𝐷 = 𝟑𝒃
𝐴𝐸 is a straight line
𝐶𝐵𝐸𝐹 is a parallelogram
𝐴𝐷: 𝐵𝐸: 𝐶𝐹 = 3: 2: 2
𝐶 𝐹
Not drawn
𝑎 accurately

𝐵 𝐸

𝐴
3𝑏 𝐷

Find expressions for the vectors below in terms of a and b.

9(a) 𝐷𝐶
[1 mark]

Answer

9(b) 𝐹𝐷
[2 marks]

Answer

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 11

9(c) DE is extended upwards until it hits the line 𝐶𝐹.

The point of intersection is 𝑋.
What is the ratio 𝐶𝑋: 𝑋𝐹?
[3 marks]

Answer

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10 On the diagram below: (Level 9)

𝐴𝐷 = 𝒂

𝐷𝐵 = 𝟑𝒃 − 𝒂
𝐴𝐶 and 𝐴𝐵 are straight lines
𝐴𝐷: 𝐷𝑌: 𝑌𝐶 = 1: 1: 1
𝐴𝑋: 𝑋𝐵 = 2: 1

Not drawn
𝑿 accurately

𝟑𝒃 − 𝒂

𝑨 𝒂 𝑫 𝒀 𝑪

Show that 𝑋𝑌 is parallel to BC.

[3 marks]

Answer

End of Questions

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