CENTRE FOR FOUNDATION STUDIES IN SCIENCE
UNIVERSITY OF MALAYA
SEM 1 SESSION 2023/2024
FAC1001: ADVANCED MATHEMATICS I
TUTORIAL 2
⃗⃗⃗⃗⃗ = 𝐛 and 𝑂𝐶
⃗⃗⃗⃗⃗ = 𝐜. P, Q, R, S and T are the
1. OABC is a rectangle with ⃗⃗⃗⃗⃗
𝑂𝐴 = 𝐚, 𝑂𝐵
⃗⃗⃗⃗⃗ , 𝑂𝐶
⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗
mid-points of ⃗⃗⃗⃗⃗
𝑂𝐴, ⃗⃗⃗⃗⃗
𝐴𝐵 , 𝐵𝐶
𝐶𝐴 respectively.
Q
A
P
B
R
C
S
⃗⃗⃗⃗⃗ , 𝑂𝑄
⃗⃗⃗⃗⃗⃗ , 𝑂𝑅
⃗⃗⃗⃗⃗ and 𝑂𝑆
⃗⃗⃗⃗⃗ in terms of a, b and c
a) Express 𝑂𝑃
b) Show that ⃗⃗⃗⃗⃗
𝑃𝑄 is parallel to ⃗⃗⃗⃗⃗
𝑆𝑅
O
⃗⃗⃗⃗⃗ = 5𝐛 and 𝑂𝐶
⃗⃗⃗⃗⃗ = 4𝐚 + 3𝐛. Show
2. O, A, B and C are four points with ⃗⃗⃗⃗⃗
𝑂𝐴 = 10𝐚, 𝑂𝐵
that A, B and C are collinear.
3.
Three points A, B and C with position vectors a, b and 3a respectively and O is the
1
origin. M is a point on AB such that ⃗⃗⃗⃗⃗⃗
𝐴𝑀 = 3 ⃗⃗⃗⃗⃗
𝐴𝐵 and N is a point on OM which has
9
been extended such that ⃗⃗⃗⃗⃗⃗
𝑂𝑁 = ⃗⃗⃗⃗⃗⃗
𝑂𝑀. Find in terms of a and b,
5
a) The position vector of N
b) Vector ⃗⃗⃗⃗⃗⃗
𝐵𝑁 and ⃗⃗⃗⃗⃗
𝑁𝐶 , hence show that B, N and C are collinear
4. In the following question, find the position vector of the point P if P divides ⃗⃗⃗⃗⃗
𝐴𝐵 in the
given ratio. Use vector method and ratio formula for comparison.
a) 𝐴(3, 5), 𝐵(−2, 3) in the ratio 3:2 externally
b) 𝐴(1, 3), 𝐵(5, 8) in the ratio -5:3
5.
Determine the resultant velocities below,
a) 20 ms-1 North and 30 ms-1 East
b) 3 ms-1 S70oE and 4 ms-1 S10oW
c) 15 ms-1 N29.3oW and 8 ms-1 N29.3oW
6. If the resultant of two coplanar forces is 15N at 30o from x-axis. Determine the other
force, if one of the forces is,
a) 20N at 0o from x-axis
b) 12N at 60o from x-axis