Cambridge International AS and A Level Mathematics Pure Mathematics 1

Exercise

(^) 8B
P1^
8
ExERCISE 8B  1  Simplify the following.
(i) 2
3

4

5







+









(ii) 2

1

1

– 2

 –





+









(iii) (^) ^34 +–– 43  (iv) 3

2

1 2

1

2







+







– 

(v) 6(3i −  2 j) − 9(2i − j)

2  The vectors p, q and r are given by

p =     3 i +   2 j + k q =     2 i +   2 j +   2 k r = − 3 i − j −     2 k.

    Find, in component form, the following vectors.

(i) p + q + r (ii) p − q (iii) p + r

(iv) 3(p − q) + 2(p + r)     (v)    4 p −   3 q +   2 r

3  In the diagram, PQRS is a parallelogram and P

→

Q = a, P

→

S = b.

(i) Write, in terms of a and b,

the following vectors.

(a) Q

→

R    (b)    P

→

R

(c) Q

→

S

(ii) The mid-point of PR is M. Find

(a) P

→

M    (b) Q

→

M.

(iii) Explain why this shows you that the

diagonals of a parallelogram bisect each other.

4  In the diagram, ABCD is a kite.

AC and BD meet at M.

    A

→

B = i + j   and 

    A

→

D = i −     2 j

(i) Use the facts that the diagonals

of a kite meet at right angles

and that M is the mid-point of

AC to find, in terms of i and j,

(a) A

→

M    (b)    A

→

C

(c) B

→

C    (d)    C

→

D.

(ii) Verify that | A

→

B | = | B

→

C | and

| A

→

D | = | C

→

D |.

Q R

P b S

a

j

i

M

A C

D

B