SPM

1. 238
CHAPTER
8
9. The position vector of city A is –10i
∼ + 10j
∼
and position vector of city B is 10i
∼ – 11j
∼
. The
position of city A, B and C are collinear and the distance between city A and city C is two
times the distance between city A and city B. The distance between cities is measured in
kilometer. Find PL4
(a) vector AB
→
,
(b) the distance between city A and city B,
(c) vector OC
→
.
10. The diagram on the right shows a quadrilateral OABC.
M is the midpoint of AC and OM : OB = 2 : 3. Given
OA
→
= 3u
∼ + 2v
∼, OC
→
= 9u
∼ + 2v
∼ and CB
→
= 3kv
∼, where k
is constant, PL4
(a) express in terms of u
∼ and/or v
∼,
(i) AC
→
(ii) OM
→
(b) express OB
→
in terms of
(i) u
∼ and v
∼, (ii) u
∼, v
∼ and k.
Hence, find the value of k.
11. The diagram on the right shows roads of a housing area that form
a rectangle OABC. Building D is at OB road and building E is at
OA road. Given OD = 3
4
OB and OE : OA = 1 : 2.
Building Y is at AB road which is extended with BY = 1
2
AB.
OA road is represented by vector 4a
∼ while OC road is
represented by vector 4c
∼. PL5
(a) Express vector that represents the following road in
terms of a
∼ and c
∼.
(i) OB
→
(ii) OD
→
(iii) OY
→
(iv) ED
→
(b) Prove that building E, D and Y reside on the same straight line.
12. The diagram on the right shows the positions and directions of
the boats which belong to Arul, Ben and Raju in a solar boat
competition. Arul’s and Ben’s boats move in the same direction of
the stream. The velocity of the stream is given by w
∼ = (i
∼ +
1
3
j
∼)m s–1
,
while the velocity of Arul’s boat is a
∼ = (3i
∼ + j
∼
) m s–1
and the
velocity of Ben’s boat is b
∼ = (6i
∼ + 2j
∼
) m s–1
. PL5
(a) Calculate the resultant velocity of Arul’s boat and the
resultant velocity of Ben’s boat. After that, find the
difference between the speed of the two boats.
(b) Raju’s boat is deviated from the path. Given that the
velocity of Raju’s boat is r
∼ = (2i
∼ –
4
3
j
∼)m s–1
. Find the unit
vector in the direction of the resultant velocity of the boat.
A
B
O E
C
Y
D
M
A
B
O
C
3u + 2v
~ ~
9u + 2v
~ ~
3kv
~
Arul Ben
Starting line
Raju