Vectors

In this question you must show detailed reasoning. Figure 4 shows a trapezium ABCD where AD is parallel to BC Given that

• AB = 2a + 3b

• BC = 15a – 5b

• DB = –4a + kb where k is an integer

show that k = 5 Given also that

• the point N lies on BC such that BN: NC = 1: 4

• AN intersects BD at X

find BX: XD

Relative to a fixed origin O,

• A is the point with position vector 12 i

• B is the point with position vector 16 j

• C is the point with position vector (50 i + 136 j)

• D is the point with position vector (22 i + 24 j)

Show that AD is parallel to BC. Points A, B, C and D are used to model the vertices of a running track in the shape of a quadrilateral. Runners complete one lap by running along all four sides of the track. The lengths of the sides are measured in metres. Given that a particular runner takes exactly 5 minutes to complete 2 laps,

calculate the average speed of this runner, giving the answer in kilometres per hour.

Relative to a fixed origin O

• the point A has position vector 4 i − 3 j + 5 k

• the point B has position vector 4 j + 6 k

• the point C has position vector −16 i + p j + 10 k

where p is a constant. Given that A, B and C lie on a straight line,

find the value of p. The line segment OB is extended to a point D so that CD is parallel to OA

Find | OD |, writing your answer as a fully simplified surd.

Figure 1 shows a sketch of triangle ABC.

Given that

find AC

show that cosABC =

Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively. Given that ● P, Q and R lie on a straight line ● Q lies one third of the way from P to R show that