Vectors 1 The diagram shows two vectors a

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

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 Q a) Write down the vector 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

4. PQRS is a trapezium. QP is parallel to RS. R S QP = 3 RS Express in terms of a and/or c (i) (ii) P Q 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

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

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.

C 9. CDEF is a quadrilateral with (a) Express D M in terms of a and b. X (b) Prove that FE is parallel to CD. so parallel Diagram NOT accurately drawn M is the midpoint of DE. (c) Express E F 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.

10. The diagram shows a regular hexagon ABCDEF with centre O. A B (a) Express in terms of a and/or b Y X (i) F C O (ii) X is the midpoint of BC. (b) Express in terms of a and/or b E D 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

Proving that the midpoints of any quadrilateral form a parallelogram using vectors C F G a B D b E H c A Therefore 2 pairs of opposite sides with the same length and direction - a parallelogram!

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

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) Q a) Write down the vector 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

4. PQRS is a trapezium. QP is parallel to RS. R S QP = 3 RS Express in terms of a and/or c (i) (ii) P Q 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

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

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 The midpoint of CB is M. (b) Prove that OPM is a straight line. C

C 9. CDEF is a quadrilateral with (a) Express D M in terms of a and b. X (b) Prove that FE is parallel to CD. M is the midpoint of DE. (c) Express E F 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. Diagram NOT accurately drawn

10. The diagram shows a regular hexagon ABCDEF with centre O. A B (a) Express in terms of a and/or b X (i) F C O (ii) X is the midpoint of BC. (b) Express in terms of a and/or b E 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 D