Cambridge International AS and A Level Mathematics Pure Mathematics 1

(Michael S) #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
Free download pdf