Vectors in geometry

7 2 26 c = 4 d = 1 Work out 2c + 3d Give your answer as a column vector.

OACB is a quadrilateral. ACP is a straight line.

M is the midpoint of OA. N is the point on BC such that BN:NC = 5: 3 → → → OA = a OB = 3b AC = 2b CP → = k AC → where k is a scalar. Given that MNP is a straight line, find the value of k. You must show all your working.

OXYZ is a parallelogram.

OY = a and OZ = b M is the point on OY such that OM:MY = 1: 3 N is the point on OZ such that ON: NZ = 1: 2 Work out the ratio XN:MN You must show all your working.

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

The diagram shows quadrilateral OACB.

M is the midpoint of OA. N is the point on BC such that BN: NC = 4: 5 OA = a OB = b AC = kbwhere k is a positive integer.

Express MN in terms of k, a and b. Give your answer in its simplest form.

Is MN parallel to OB? Give a reason for your answer.

On the grid below, draw and label the vector 2a + b

ABCD is a quadrilateral. E, F, G and H are the midpoints of AB, BC, CD and DA. → → → AH = a AE = b DG = c Prove, using vectors, that EFGH is a parallelogram.

Enlarge triangle P by scale factor –1 with centre of enlargement (–2, –1) Label your image Q. Triangle P is transformed by a combined transformation of a rotation of 90° anticlockwise about the origin followed by a translation to give triangle R. Exactly one vertex of triangle P is invariant under the combined transformation.

Find one possible column vector for the translation.

ORT is a triangle.

OT→ = a RT→ = b M is the point on OR such that OM:MR = 2: 3 Express MT→ in terms of a and b. Give your answer in its simplest form.

2 8 13 22Given that the vector a b is parallel to the vector

find an expression for b in terms of a.

Work out as a column vector

a + b

2a − c

The vector d is drawn on the grid.

From the point P, draw the vector 2d

CDEF is a quadrilateral.

The point P is such that CEP is a straight line and that CE = EP Use a vector method to prove that CF is parallel to DP.

A, B and C are three points such that AB = 3a + 4b AC = 15a + 20b

Prove that A, B and C lie on a straight line.

D, E and F are three points on a straight line such that DE = 3e + 6f EF = −10.5e − 21f

Find the ratio length of DF: length of DE

Describe fully the single transformation that maps shape S onto shape T.

Describe fully the single transformation that maps shape S onto shape T.

a and b are vectors such that

Find b as a column vector.

OABC is a trapezium.

OA → = a AB → = b OC → = 3b D is the point on OB such that OD:DB = 2: 3 E is the point on BC such that BE: EC = 1: 4 Work out the vector DE → in terms of a and b. Give your answer in its simplest form.

DEF is a triangle.

P is the midpoint of FD. Q is the midpoint of DE. FD= a and FE = b Use a vector method to prove that PQ is parallel to FE.

Find 2a – 3b as a column vector.

Find 2a – 3b as a column vector.

OXYZ is a parallelogram.

P is the point on OX such that OP: PX = 1: 2 R is the point on OY such that OR: RY = 1: 3

Work out, in its simplest form, the ratio ZP: ZR You must show all your working.

Shape A can be transformed to shape B by a reflection in the x-axis followed by a

Find the value of c and the value of d.

Shape A can be transformed to shape B by a reflection in the x-axis followed by a

Find the value of c and the value of d.

Work out ab as a column vector.

CDEF is a quadrilateral.

CD = a, DE = b and FC = a – b.

Express FE in terms of a and/or b. Give your answer in its simplest form.

M is the midpoint of DE. X is the point on FM such that FX: XM = n: 1 CXE is a straight line.

Work out the value of n.

OAB is a triangle. OPM and APN are straight lines. M is the midpoint of AB. OA = a OB = b

OP: PM = 3: 2

Work out the ratio ON: NB

Work out 2 a + b as a column vector.

The vector a and the vector b are shown on the grid.

On the grid, draw and label vector –2 a

Work out a + 2 b as a column vector.

Rotate trapezium T 180° about the origin. Label the new trapezium A.

Rotate trapezium T 180ƒ about the origin. Label the new trapezium A.

OAN, OMB and APB are straight lines. AN = 2 OA. M is the midpoint of OB. OA = a OB = b AP = k AB where k is a scalar quantity.

Given that MPN is a straight line, find the value of k.

ABCD is a parallelogram. The diagonals of the parallelogram intersect at O. OA = a and OB = b

Find, in terms of b, the vector DB.

Find, in terms of a and b, the vector AB.

Find, in terms of a and b, the vector AD.