equation of line_vectors

IB Questionbank Mathematics Higher Level 3rd edition 1

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

6

7

4

,

3

4

,

2

3

2

cba

x

y

x

x

x

y

where 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)

2. (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 containing the vectors a and b and passing through the point (1, 1, 1).

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

(11)

(b) Given two vectors p and q,

(i) show that p p = p2;

(ii) hence, or otherwise, show that p + q2 = p2 + 2p q + q2;

(iii) deduce that p + qp + q. (8)

(Total 19 marks)

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

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


(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

2

2

1

1OQ

t

t

t

, t 0.

(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)

4. 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 that contains the face ABC. (4)

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

(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)

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

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

(b) The vector product PSPQ

m

7

13

. Find the value of m.

(2)

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

(d) Find the Cartesian equation of the plane, 1, containing the parallelogram PQRS. (3)

(e) Write down the vector equation of the line through the origin (0, 0, 0) that is


perpendicular to the plane 1.

(1)

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

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

planes. (4)

(Total 17 marks)

6. (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 1

3

2

2

1

1

zyx. Show that the lines

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, , 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 and passes

through the point T(3, 1, 4). (2)

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

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


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

(Total 22 marks)

7. Find the angle between the lines 2

1x = 1 y = 2z and x = y = 3z.

(Total 6 marks)

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

(a) Calculate the cosine of the angle between OA and 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 of

intersection. (7)

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

(6)

(Total 20 marks)


9. (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 that contains these two lines. (4)

(d) Let B be the point of intersection of the plane 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 plane

and passing through C. (3)

(Total 18 marks)

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

(Total 6 marks)

equation of line_vectors

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