Vectors in geometry

OAB is a triangle. P is the midpoint of OA Q is a point on OB

ABR and PQR are straight lines. OA→ = 12a OB→ = 8b

Express AB→ in terms of a and b

AB:BR = 1: 2 OQ→ = nb

Use a vector method to find the value of n

22

OACB is a quadrilateral.

Find the vector in terms of a and b

The point P lies on AB and on OC

Using a vector method, find the ratio AP: PB Show your working clearly.

OMA, ONB, MPB and NPA are straight lines.

M is the midpoint of OA

ON:NB = 1: 5

OM → = a ON → = b

Find in terms of a and b the vector AN →

Use a vector method to find the ratio AP:PN

AP:PN =_........

The diagram shows a triangle ABC where A, B and C represent the positions of three towns.

Pru travels directly from A to B and then directly from B to C

Yang travels directly from A to C Given that the values for AB → and BC → are in kilometres,

work out how much further Pru travels than Yang travels. Give your answer in km, correct to one decimal place.

The diagram shows triangle OAB with OA extended to E

OA = a OB = b

M is the point on OB such that OM:MB = 4: 1 N is the point on AB such that AN: NB = 3: 2 OA: AE = 5: 3

Find an expression for ON →in terms of a and b Give your answer in its simplest form.

ON → =_........

Use a vector method to show that MNE is a straight line.

OAED is a quadrilateral.

AB:BD = 1: 2

The point C on AB is such that OCE is a straight line.

Use a vector method to find the ratio of OC:CE

Here are two vectors.

Find AC → as a column vector.

Describe fully the single transformation that maps triangle A onto triangle B

Label your triangle C

ABCD is a parallelogram and ADM is a straight line.

Diagram NOT

K is the point on AB such that AK:AB = λ: 1 L is the point on CD such that CL: CD = μ: 1 KLM is a straight line.

Given that λ:μ = 1: 2

use a vector method to find the value of λ and the value of μ

The diagram shows triangle OAB

OA → = 8a OB → = 6b

M is the point on OB such that OM:MB = 1: 2 N is the midpoint of AB P is the point of intersection of ON and AM Using a vector method, find → OP as a simplified expression in terms of a and b Show your working clearly.

ABCDEF and GHIJKL are regular hexagons each with centre O.

GHIJKL is an enlargement of ABCDEF, with centre O and scale factor 2 OA = a OB = b

Write the following vectors, in terms of a and b. Simplify your answers.

AB

KI

LD

The triangle OAB has an area of 5 cm²

Calculate the area of the shaded region.

Here are two vectors.

Find, as a column vector, → AC

OAB is a triangle.

OA → = 2a and OB → = 2b

M is the midpoint of AB. N is the point on OB such that ON: NB = 2: 1

P is the point on AN such that OPM is a straight line.

Use a vector method to find OP:PM Show your working clearly.

Here is a shape P drawn on a grid of squares.

On the grid, rotate shape P 180 about the point (–3, 2) Label the new shape Q.

Here are triangle T and triangle U drawn on a grid of squares.

Describe fully the single transformation that maps triangle T onto triangle U.

OAB is a triangle. OA = a OB = b

The point C lies on OA such that OC: CA = 1: 2 The point D lies on OB such that OD: DB = 1: 2

Using a vector method, prove that ABDC is a trapezium.

The diagram shows trapezium OACB.

OA = 3a OB = 6b AC = 4b

N is the point on OC such that ANB is a straight line. Find ON as a simplified expression in terms of a and b.

OA = a OC = c AB = 2c

P is the point on AB such that AP: PB = 3: 1 Q is the point on AC such that OQP is a straight line.

Use a vector method to find AQ: QC Show your working clearly.

Here are two vectors.

Find the magnitude of AC.

The diagram shows parallelogram ABCD.

The point B has coordinates (5, 8)

Work out the coordinates of the point C.

The point E has coordinates (63, 211)

Use a vector method to prove that ABE is a straight line.

OAB is a triangle. OA = a OB = b

C is the midpoint of OA. D is the point on AB such that AD: DB = 3: 1 E is the point such that OB 2BE

Using a vector method, prove that the points C, D and E lie on the same straight line.