Vectors
(Level 6 - 9)
2
1 ๐พ๐ฟ๐ is a scalene triangle. (Level 6)
Point P is half way along ๐๐ฟ.
๐๐พ = ๐๐, ๐พ๐ฟ = ๐.
1(a) Write the vector ๐๐ฟ in terms of ๐ and ๐.
[1 mark]
Answer
1(b) Find ๐๐ in terms of ๐ and ๐.
[1 mark]
Answer
1(c) A new point ๐ is to the right of ๐ฟ.
๐๐ is 4 times the length of ๐๐.
Write ๐๐ as a column vector.
[1 mark]
Answer
Turn over โบ
3
Not drawn
accurately
๐ ๐ฟ
๐พ
2๐
๐
๐
3
2 ๐ธ๐น๐บ is a scalene triangle. (Level 6)
๐ธ๐บ = ๐ โ 5๐, ๐ธ๐น = 3๐ + 4๐.
๐บ๐ป, not shown, is 3 times the length of and parallel to ๐บ๐น.
Find ๐บ๐ป in terms of ๐ and ๐.
[3 marks]
Answer
Turn over โบ
3
E
G
F
๐ โ 5๐
3๐ + 4๐
Not drawn
accurately
4
3 ๐ด๐ต๐ถ๐ท is a rhombus. Opposite sides are parallel. (Level 7)
๐ต๐ธ is an extension of ๐ด๐ต, such that ๐ด๐ต โถ ๐ต๐ธ = 4: 3
The point F is halfway along ๐ถ๐ท
Find ๐น๐ธ in terms of ๐ and ๐.
[3 marks]
Answer
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Turn over โบ
3
๐ต ๐ธ
๐ถ
a
b
๐ด
๐ท ๐น
Not drawn
accurately
5
4 KLMN is an irregular quadrilateral. (Level 7)
๐๐ฟ = ๐๐, ๐๐ = 2๐ + ๐๐, ๐๐พ = 3๐ + ๐.
Point P is positioned such that ๐ฟ๐: ๐๐พ = 1: 2
4(a) Find the vector ๐ฟ๐พ in terms of ๐ and ๐.
[2 marks]
Answer
4(b) Show that ๐๐ = ๐๐พ.
[4 marks]
Answer
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6
๐L
K
3๐ + ๐
2๐ + 2๐P
๐
2๐
Not drawn
accurately
6
5 The diagram below shows a regular hexagon ABCDEF. (Level 7)
X is the point at the center of the hexagon.
๐๐ต = ๐
๐ธ๐ท = ๐
Write down the following vectors in terms of a and b.
5(a) ๐ต๐ถ
[1 mark]
5(b) ๐ต๐ธ
[1 mark]
5(c) ๐ด๐ธ
[1 mark]
5(d) ๐น๐ต
[1 mark]
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4
๐ต
๐ด
๐ถ
๐ท
๐ธ
๐น
๐
๐
๐
Not drawn
accurately
7
6 ๐ด๐ต๐ถ is an isosceles triangle. (Level 7)
Point ๐ is halfway along the triangle base, ๐ด๐ถ.
Point ๐ท is positioned such that ๐ด๐ถ and ๐ด๐ท are collinear.
Point ๐ธ is positioned such that ๐ด๐ต and ๐ด๐ธ are collinear.
๐ด๐ต:๐ต๐ธ = 3:2
๐๐ด = ๐, ๐ด๐ต = 3๐, ๐ด๐ท = 2๐ด๐ถ.
Find ๐ท๐ธ in terms of ๐ and ๐.
[4 marks]
Answer
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4
๐ด ๐ถ
๐ต
b
3a
Not drawn
accurately
M ๐ท
E
8
7 On the diagram below: (Level 7)
AE is a straight line
๐ด๐ต = ๐
๐ด๐ถ = ๐
๐ด๐ถ = ๐ถ๐ธ
๐ท๐ธ =1
4๐ถ๐ธ
Find expressions for the vectors below in terms of a and b.
7(a) ๐ต๐ถ
[1 mark]
Answer
7(b) ๐ท๐ต
[4 marks]
Answer
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5
๐ด
๐ต
๐ถ ๐ท ๐ธ๐
๐
Not drawn
accurately
9
8 ๐ด๐ต๐ถ is a scalene triangle. (Level 8)
๐ด๐ถ = 3๐, ๐ด๐ต = 2๐.
Point D is positioned such that ๐ต๐ท:๐ท๐ถ = 1: 3
Point E is positioned such that ๐ด๐ธ:๐ธ๐ท = 1: 1
Find ๐ด๐ธ in terms of ๐ and ๐.
[5 marks]
Answer
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5
A C
B
3b
2a
D
E
Not drawn
accurately
10
9 On the diagram below: (Level 8)
๐ด๐ต = ๐ต๐ถ = ๐
๐ด๐ท = ๐๐
๐ด๐ธ is a straight line
๐ถ๐ต๐ธ๐น is a parallelogram
๐ด๐ท: ๐ต๐ธ: ๐ถ๐น = 3: 2: 2
Find expressions for the vectors below in terms of a and b.
9(a) ๐ท๐ถ
[1 mark]
Answer
9(b) ๐น๐ท
[2 marks]
Answer
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3
๐ด
๐ต
๐ท3๐
๐
๐ถ
๐ธ
๐น
๐Not drawn
accurately
11
9(c) DE is extended upwards until it hits the line ๐ถ๐น.
The point of intersection is ๐.
What is the ratio ๐ถ๐: ๐๐น?
[3 marks]
Answer
Turn over โบ
3
12
10 On the diagram below: (Level 9)
๐ด๐ท = ๐
๐ท๐ต = ๐๐ โ ๐
๐ด๐ถ and ๐ด๐ต are straight lines
๐ด๐ท:๐ท๐: ๐๐ถ = 1: 1: 1
๐ด๐: ๐๐ต = 2: 1
Show that ๐๐ is parallel to BC.
[3 marks]
Answer
End of Questions
END
3
๐ฉ
๐จ ๐ช๐๐ซ
๐ฟ
๐
๐๐ โ ๐
Not drawn
accurately