1. The diagram below shows a circle with centre O. The points A, B, C lie on the circumference ofthe circle and [AC] is a diameter.

Let .ba OB and OA

(a) Write down expressions for in terms of the vectors a and b.CB and AB(2)

(b) Hence prove that angle is a right angle.CBA(3)

(Total 5 marks)

2. The points A(1, 2, 1), B(–3, 1, 4), C(5, –1, 2) and D(5, 3, 7) are the vertices of a tetrahedron.

(a) Find the vectors .AC and AB(2)

(b) Find the Cartesian equation of the plane Π that contains the face ABC.(4)

(c) Find the vector equation of the line that passes through D and is perpendicular to Π.Hence, or otherwise, calculate the shortest distance to D from Π.

(5)

(d) (i) Calculate the area of the triangle ABC.

(ii) Calculate the volume of the tetrahedron ABCD.(4)

(e) Determine which of the vertices B or D is closer to its opposite face.(4)

(Total 19 marks)

3. In the diagram below, [AB] is a diameter of the circle with centre O. Point C is on the

circumference of the circle. Let .cb OC and OB

(a) Find an expression for and for in terms of b and c.CB AC(2)

(b) Hence prove that is a right angle.BCA(3)

(Total 5 marks)

4. Port A is defined to be the origin of a set of coordinate axes and port B is located at the point(70, 30), where distances are measured in kilometres. A ship S1 sails from port A at 10:00 in a

straight line such that its position t hours after 10:00 is given by r =

.

20

10t

A speedboat S2 is capable of three times the speed of S1 and is to meet S1 by travelling theshortest possible distance. What is the latest time that S2 can leave port B?

(Total 7 marks)

5. The equations of three planes, are given by

ax + 2y + z = 3 –x + (a + 1)y + 3z = 1 –2x + y + (a + 2)z = k

where a .

(a) Given that a = 0, show that the three planes intersect at a point.(3)

(b) Find the value of a such that the three planes do not meet at a point.(5)

(c) Given a such that the three planes do not meet at a point, find the value of k such that theplanes meet in one line and find an equation of this line in the form

.

n

m

l

z

y

x

z

y

x

0

0

0

(6)(Total 14 marks)

6. Consider the functions f(x) = x3 + 1 and g(x) =

. The graphs of y = f(x) and y = g(x) meet1

13 x

at the point (0, 1) and one other point, P.

(a) Find the coordinates of P.(1)

(b) Calculate the size of the acute angle between the tangents to the two graphs at thepoint P.

(4)(Total 5 marks)

7. The points P(–1, 2, –3), Q(–2, 1, 0), R(0, 5, 1) and S form a parallelogram, where S isdiagonally opposite Q.

(a) Find the coordinates of S.(2)

(b) The vector product

. Find the value of m. PSPQ

m

7

13

(2)

(c) Hence calculate the area of parallelogram PQRS.(2)

(d) Find the Cartesian equation of the plane, Π1, containing the parallelogram PQRS.(3)

(e) Write down the vector equation of the line through the origin (0, 0, 0) that isperpendicular to the plane Π1.

(1)

(f) Hence find the point on the plane that is closest to the origin.(3)

(g) A second plane, Π2, has equation x – 2y + z = 3. Calculate the angle between the twoplanes.

(4)(Total 17 marks)

8. (a) Show that the two planes

π1 : x + 2y – z = 1π2 : x + z = –2

are perpendicular.(3)

(b) Find the equation of the plane π3 that passes through the origin and is perpendicular toboth π1 and π2.

(4)(Total 7 marks)

9. Consider the vectors . Show that if │a│=│b│ thenbaba OC and OB,OA

(a + b)•(a – b) = 0. Comment on what this tells us about the parallelogram OACB.(Total 4 marks)

10. The three vectors a, b and c are given by

where x, y .

6

7

4

,

3

4

,

2

3

2

cba

x

y

x

x

x

y

(a) If a + 2b – c = 0, find the value of x and of y.(3)

(b) Find the exact value of │a + 2b│.(2)

(Total 5 marks)

11. (a) Consider the vectors a = 6i + 3j + 2k, b = –3j + 4k.

(i) Find the cosine of the angle between vectors a and b.

(ii) Find a × b.

(iii) Hence find the Cartesian equation of the plane Π containing the vectors a and band passing through the point (1, 1, –1).

(iv) The plane Π intersects the x-y plane in the line l. Find the area of the finitetriangular region enclosed by l, the x-axis and the y-axis.

(11)

(b) Given two vectors p and q,

(i) show that p • p = │p│2;

(ii) hence, or otherwise, show that │p + q│2 = │p│2 + 2p • q + │q│2;

(iii) deduce that │p + q│≤│p│ + │q│.(8)

(Total 19 marks)

12. The system of equations

2x – y + 3z = 23x + y + 2z = –2–x + 2y + az = b

is known to have more than one solution. Find the value of a and the value of b.(Total 5 marks)

13. A plane π has vector equation r = (–2i + 3j – 2k) + λ(2i + 3j + 2k) + μ(6i – 3j + 2k).

(a) Show that the Cartesian equation of the plane π is 3x + 2y – 6z = 12.(6)

(b) The plane π meets the x, y and z axes at A, B and C respectively. Find the coordinates ofA, B and C.

(3)

(c) Find the volume of the pyramid OABC.(3)

(d) Find the angle between the plane π and the x-axis.(4)

(e) Hence, or otherwise, find the distance from the origin to the plane π.(2)

(f) Using your answers from (c) and (e), find the area of the triangle ABC.(2)

(Total 20 marks)

14. The three planes

2x – 2y – z = 34x + 5y – 2z = –33x + 4y – 3z = –7

intersect at the point with coordinates (a, b, c).

(a) Find the value of each of a, b and c.(2)

(b) The equations of three other planes are

2x – 4y – 3z = 4–x + 3y + 5z = –23x – 5y – z = 6.

Find a vector equation of the line of intersection of these three planes.(4)

(Total 6 marks)

15. Consider the plane with equation 4x – 2y – z = 1 and the line given by the parametric equations

x = 3 – 2λ y = (2k – 1) + λ z = –1 + kλ.

Given that the line is perpendicular to the plane, find

(a) the value of k;(4)

(b) the coordinates of the point of intersection of the line and the plane.(4)

(Total 8 marks)

16. Consider the vectors a = sin(2α)i – cos(2α)j + k and b = cos α i – sin α j – k, where 0 < α < 2π.

Let θ be the angle between the vectors a and b.

(a) Express cos θ in terms of α.(2)

(b) Find the acute angle α for which the two vectors are perpendicular.(2)

(c) For α =

, determine the vector product of a and b and comment on the geometrical6

π7

significance of this result.(4)

(Total 8 marks)

17. The diagram shows a cube OABCDEFG.

Let O be the origin, (OA) the x-axis, (OC) the y-axis and (OD) the z-axis.Let M, N and P be the midpoints of [FG], [DG] and [CG], respectively.The coordinates of F are (2, 2, 2).

(a) Find the position vectors in component form.OP and ON,OM

(3)

(b) Find .MNMP(4)

(c) Hence,

(i) calculate the area of the triangle MNP;

(ii) show that the line (AG) is perpendicular to the plane MNP;

(iii) find the equation of the plane MNP.(7)

(d) Determine the coordinates of the point where the line (AG) meets the plane MNP.(6)

(Total 20 marks)

18. A triangle has vertices A(1, –1, 1), B(1, 1, 0) and C(–1, 1, –1).

Show that the area of the triangle is .6

(Total 6 marks)

19. (a) Show that a Cartesian equation of the line, l1, containing points A(1, –1, 2) and B(3, 0, 3)

has the form

.1

2

1

1

2

1

zyx

(2)

(b) An equation of a second line, l2, has the form

. Show that the lines1

3

2

2

1

1

zyx

l1 and l2 intersect, and find the coordinates of their point of intersection.(5)

(c) Given that direction vectors of l1 and l2 are d1 and d2 respectively, determine d1 × d2.(3)

(d) Show that a Cartesian equation of the plane, Π, that contains l1 and l2 is –x – y + 3z = 6.(3)

(e) Find a vector equation of the line l3 which is perpendicular to the plane Π and passesthrough the point T(3, 1, –4).

(2)

(f) (i) Find the point of intersection of the line l3 and the plane Π.

(ii) Find the coordinates of T′, the reflection of the point T in the plane Π.

(iii) Hence find the magnitude of the vector .TT (7)

(Total 22 marks)

20. Find the angle between the lines

= 1 – y = 2z and x = y = 3z.2

1x

(Total 6 marks)

21. Consider the planes defined by the equations x + y + 2z = 2, 2x – y + 3z = 2 and5x – y + az = 5 where a is a real number.

(a) If a = 4 find the coordinates of the point of intersection of the three planes.(2)

(b) (i) Find the value of a for which the planes do not meet at a unique point.

(ii) For this value of a show that the three planes do not have any common point.(6)

(Total 8 marks)

22. The position vector at time t of a point P is given by

= (1 + t)i + (2 – 2t)j + (3t – 1)k, t ≥ 0.OP

(a) Find the coordinates of P when t = 0.(2)

(b) Show that P moves along the line L with Cartesian equations

x – 1 =

.3

1

2

2

zy

(2)

(c) (i) Find the value of t when P lies on the plane with equation 2x + y + z = 6.

(ii) State the coordinates of P at this time.

(iii) Hence find the total distance travelled by P before it meets the plane.(6)

The position vector at time t of another point, Q, is given by

, t ≥ 0.

2

2

1

1OQ

t

t

t

(d) (i) Find the value of t for which the distance from Q to the origin is minimum.

(ii) Find the coordinates of Q at this time.(6)

(e) Let a, b and c be the position vectors of Q at times t = 0, t = 1, and t = 2 respectively.

(i) Show that the equation a – b = k(b – c) has no solution for k.

(ii) Hence show that the path of Q is not a straight line.(7)

(Total 23 marks)

23. Given that a = 2 sin θ i + (1 – sin θ)j, find the value of the acute angle θ, so that a isperpendicular to the line x + y = 1.

(Total 5 marks)

24. The vector equation of line l is given as

.

1

2

1

6

3

1

z

y

x

Find the Cartesian equation of the plane containing the line l and the point A(4, –2, 5).(Total 6 marks)

25. Two lines are defined by

l1 : r =

= –(z + 3).4

7

3

4: and

2

2

3

6

4

3

2

yxl

(a) Find the coordinates of the point A on l1 and the point B on l2 such that isABperpendicular to both l1 and l2.

(13)

(b) Find │AB│.(3)

(c) Find the Cartesian equation of the plane Π that contains l1 and does not intersect l2.(3)

(Total 19 marks)

26. The points A, B, C have position vectors i + j + 2k, i + 2 j + 3k, 3i + k respectively and lie inthe plane .

(a) Find

(i) the area of the triangle ABC;

(ii) the shortest distance from C to the line AB;

(iii) the cartesian equation of the plane .(14)

The line L passes through the origin and is normal to the plane , it intersects at the point D.

(b) Find

(i) the coordinates of the point D;

(ii) the distance of from the origin.(6)

(Total 20 marks)

27. Given any two non-zero vectors a and b, show that │a × b│2 = │a│2│b│2 – (a • b)2.(Total 6 marks)

28. Consider the points A(1, −1, 4), B(2, − 2, 5) and O(0, 0, 0).

(a) Calculate the cosine of the angle between and OA AB.(5)

(b) Find a vector equation of the line L1 which passes through A and B.(2)

The line L2 has equation r = 2i + 4j + 7k + t(2i + j + 3k), where t .

(c) Show that the lines L1 and L2 intersect and find the coordinates of their point ofintersection.

(7)

(d) Find the Cartesian equation of the plane that contains both the line L2 and the point A.(6)

(Total 20 marks)

29. A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector

and

2

1

4

meets the plane π : x + 3y + 2z − 24 = 0.

Find the angle that the ray of light makes with the plane.(Total 6 marks)

30. Find the vector equation of the line of intersection of the three planes represented by thefollowing system of equations.

2x − 7y + 5z =1 6x + 3y – z = –1 −14x − 23y +13z = 5

(Total 6 marks)

31. Three distinct non-zero vectors are given by .cba OC and,OB,OA

If is perpendicular to and is perpendicular to , show that isOA BC OB CA OC

perpendicular to .AB(Total 6 marks)

32. The angle between the vector a = i – 2j + 3k and the vector b = 3i – 2j + mk is 30°.

Find the values of m.(Total 6 marks)

33. (a) Find the set of values of k for which the following system of equations has no solution.

x + 2y – 3z = k 3x + y + 2z = 4 5x + 7z = 5

(4)

(b) Describe the geometrical relationship of the three planes represented by this system ofequations.

(1)(Total 5 marks)

34. (a) Write the vector equations of the following lines in parametric form.

r1 =

2

1

2

7

2

3

m

r2 =

1

1

4

2

4

1

n

(2)

(b) Hence show that these two lines intersect and find the point of intersection, A.(5)

(c) Find the Cartesian equation of the plane Π that contains these two lines.(4)

(d) Let B be the point of intersection of the plane Π and the line r =

.

2

8

3

0

3

8

Find the coordinates of B.(4)

(e) If C is the mid-point of AB, find the vector equation of the line perpendicular to the planeΠ and passing through C.

(3)(Total 18 marks)

35. The line L is given by the parametric equations x = 1 – λ, y = 2 – 3λ, z = 2.Find the coordinates of the point on L that is nearest to the origin.

(Total 6 marks)

36. (a) Show that the following system of equations will have a unique solution when a ≠ –1.

x + 3y – z = 0 3x + 5y – z = 0

x – 5y + (2 – a)z = 9 – a2

(5)

(b) State the solution in terms of a.(6)

(c) Hence, solve

x + 3y – z = 0 3x + 5y – z = 0 x – 5y + z = 8

(2)(Total 13 marks)

37. Consider the points A(1, 2, 1), B(0, –1, 2), C(1, 0, 2) and D(2, –1, –6).

(a) Find the vectors and .AB BC(2)

(b) Calculate .BCAB(2)

(c) Hence, or otherwise find the area of triangle ABC.(3)

(d) Find the Cartesian equation of the plane P containing the points A, B and C.(3)

(e) Find a set of parametric equations for the line L through the point D and perpendicular tothe plane P.

(3)

(f) Find the point of intersection E, of the line L and the plane P.(4)

(g) Find the distance from the point D to the plane P.(2)

(h) Find a unit vector that is perpendicular to the plane P.(2)

(i) The point F is a reflection of D in the plane P. Find the coordinates of F.(4)

(Total 25 marks)

38. (a) Show that lines

and

intersect and find the1

3

3

2

1

2

zyx

2

4

4

3

1

2

zyx

coordinates of P, the point of intersection.(8)

(b) Find the Cartesian equation of the plane Π that contains the two lines.(6)

(c) The point Q(3, 4, 3) lies on Π. The line L passes through the midpoint of [PQ]. Point S is

on L such that

= 3, and the triangle PQS is normal to the plane Π. Given thatQSPS

there are two possible positions for S, find their coordinates.(15)

(Total 29 marks)

39. A triangle has its vertices at A(–1, 3, 2), B(3, 6, 1) and C(–4, 4, 3).

(a) Show that = –10.ACAB(3)

(b) Find .CAB

(5)(Total 8 marks)