1. Vectors 1. The diagram shows two vectors a and b.
On the grid above draw the vector

2. 2.
Write down as a column vector
(i)
(ii) R 3.
The diagram is a sketch.
P is the point (2, 3) Q is the point (6, 6) S
a) Write down the vector Q
Write your answer as a column vector P
R is above Q
Diagram NOT accurately drawn
PQRS is a parallelogram.
(b) Find the vector
Write your answer as a column vector

3. 4. PQRS is a trapezium. QP is parallel to RS.
QP = 3RS
Express in terms of a and/or c P Q
(i)
(ii)
Diagrams NOT accurately drawn A
5. OAB is a triangle.
P is the point on AB such that AP : PB = 2 : 1
Write in terms of a and b P O B

4. 6. OPQ is a triangle. Q
T is the point on PQ for which PT : TQ = 2 : 1 T
(a) Write down, in terms of a and b,
an expression for O P
(b) Express in terms of a and b.
Give your answer in its simplest form.
Diagrams NOT accurately drawn X
7. OABC is a trapezium. OC is parallel to AB. B A
AB = 2OC
X is the point on AB such that AX:XB = 3:1.
Express in terms of a and c. O C

5. 8. OABC is a parallelogram.
P is the point on AC such that AP = AC A B
(a) Find the vector
Give your answer in terms of a and c M P O C
The midpoint of CB is M.
(b) Prove that OPM is a straight line.
As parallelogram,
and OPM is a straight line.

6. 9. CDEF is a quadrilateral with
(a) Express in terms of a and b. M X
(b) Prove that FE is parallel to CD. F E
so parallel
Diagram NOT accurately drawn
M is the midpoint of DE.
(c) Express in terms of a and b.
X is the point on FM such that FX : XM = 4 : 1
(d) Prove that C, X and E lie on the same straight line.
and CXE is a straight line.

7. 10.The diagram shows a regular hexagon ABCDEF with centre O.Y
(a) Express in terms of a and/or b X
(i) F C O
(ii)
X is the midpoint of BC. E D
(b) Express
in terms of a and/or b
Y is the point on AB extended, such that AB : BY = 3:2
(c) Prove that E, X and Y lie on the same straight line
and EXY is a straight line.
A regular hexagon can be split into 6 equilateral triangles

8. Proving that the midpoints of any quadrilateral form a parallelogram using vectorsB D b E H c A
Therefore 2 pairs of opposite sides with the same length and direction - a parallelogram!

9. Vectors 1. The diagram shows two vectors a and b.
On the grid above draw the vector

10. 2.
Write down as a column vector
(i)
(ii) 3.
The diagram is a sketch.
P is the point (2, 3) Q is the point (6, 6)
a) Write down the vector Q
Write your answer as a column vector P
Diagram NOT accurately drawn
PQRS is a parallelogram.
(b) Find the vector
Write your answer as a column vector

11. 4. PQRS is a trapezium. QP is parallel to RS.
QP = 3RS
Express in terms of a and/or c P Q
(i)
(ii)
Diagrams NOT accurately drawn A
5. OAB is a triangle.
P is the point on AB such that AP : PB = 2 : 1
Write in terms of a and b P O B

12. 6. OPQ is a triangle. Q
T is the point on PQ for which PT : TQ = 2 : 1 T
(a) Write down, in terms of a and b,
an expression for O P
(b) Express in terms of a and b.
Give your answer in its simplest form.
Diagrams NOT accurately drawn X
7. OABC is a trapezium. OC is parallel to AB. B A
AB = 2OC
X is the point on AB such that AX:XB = 3:1.
Express in terms of a and c. O C

13. 8. OABC is a parallelogram.
P is the point on AC such that AP = AC A B
(a) Find the vector
Give your answer in terms of a and c P O C
The midpoint of CB is M.
(b) Prove that OPM is a straight line.

14. 9. CDEF is a quadrilateral with
(a) Express in terms of a and b. M X
(b) Prove that FE is parallel to CD. F E
Diagram NOT accurately drawn
M is the midpoint of DE.
(c) Express in terms of a and b.
X is the point on FM such that FX : XM = 4 : 1
(d) Prove that C, X and E lie on the same straight line.

15. 10.The diagram shows a regular hexagon ABCDEF with centre O.
(a) Express in terms of a and/or b X
(i) F C O
(ii)
X is the midpoint of BC. E D
(b) Express
in terms of a and/or b
Y is the point on AB extended, such that AB : BY = 3:2
(c) Prove that E, X and Y lie on the same straight line