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Math 111: Vector Geometry Question Paper
Math 111: Vector Geometry
Course:Bachelor Of Computer Science (It Telecommunication)
Institution: Kabarak University question papers
Exam Year:2011
KABARAK UNIVERSITY
UNIVERSITY EXAMINATIONS
2011/2012 ACADEMIC YEAR
FOR THE DEGREE OF BACHELOR OF COMPUTER SCIENCE
MATH 111: VECTOR GEOMETRY
DAY: THURSDAY DATE: 29/03/2012
TIME: 2.00 – 4.00 P.M. STREAM: Y1S1
INSTRUCTIONS:
1.Question ONE is compulsory.
2. Attempt question ONE and any other TWO
Question One [30 Marks]
a) Distinguish between the following. [3 Marks]
i. Scalar and Vector quantity
ii. Free vector and position vector
iii. Two dimensions and three dimensions
b) ABCD is a quadrilateral in which P and Q are the mid points of the diagonals AC and
BD respectively. Show that AB + AD + CB + CD = 4PQ [4 Marks]
c) Find the magnitude and direction of the displacement vector PQ where P and Q are
points (9,7)and (12,4) respectively. [4 Marks]
d) Find the area of a triangle whose vertices are P(1,3,2) Q(2,-1,1) R(-1,2,3)
[3 Marks]
e) Determine a unit vector that is perpendicular to the plane of A = 2iˆ - 6 ˆj - 3kˆ
r
and B = 4iˆ + 3 ˆj - kˆ
r
[4 marks]
f) Find the angles which the vector A = 3iˆ - 6 ˆj + 2kˆ
r
makes with the coordinate axis.
[4 Marks]
g) The centroid of triangle OAB is denoted by G. If O is the origin and
j i OA ˆ 3 ˆ 4 + = j i OB ˆ ˆ 6 - = findOG in terms of the unit vectors iˆ and jˆ [3 Marks]
h) Determine the value of a such that the vectors p = 2iˆ + aˆj + 4kˆ and
q = 5iˆ + 2 ˆj - 4kˆ are perpendicular. [3 Marks]
Question Two [20 Marks]
a) Evaluate (2iˆ-3ˆj)•(iˆ+ ˆj -kˆ)×(3iˆ -kˆ). [4 Marks]
b) Find the resultant of the following displacements A, 10m northwest; B, 20m 300
north of east; C, 35m due south. [6 marks]
c) the vector equations of three lines are given below
L : r 17iˆ 2 ˆj 6kˆ ( 9iˆ 3 ˆj 9kˆ) 1 = + - + l - + +
L : r 2iˆ 3 ˆj 4kˆ (6iˆ 7 ˆj kˆ) 2 = - + + µ + -
L : r 2iˆ 12 ˆj kˆ ( 3iˆ ˆj 3kˆ ) 3 = - - +h - + +
State which of the lines are [6 marks]
i. Parallel
ii. Intersect
iii. Are skewed
d) Show that addition of vectors is associative. [4 marks]
Question Three [20 Marks]
a) Find the work done in moving an object along a vector r = 3iˆ + 2 ˆj - 5kˆ if the applied
force is F = 2iˆ - ˆj - kˆ [3 Marks]
b) Find the perpendicular distance between the point A(4, - 3,10) and the line L whose
vector equation is
2
1
3
3
2
1
r l [6 Marks]
c) A stationery observer O observes a ship S at noon at a point whose coordinates
relative to O are (20,15); units are in kilometers. The ship is moving at a steady speed
of 10 km/h on a bearing 1500.
i) Express its velocity as a column vector. [3 Marks]
ii) Write down in terms of t, its position after t hours [3 Marks]
iii) Find the value of t when the ship is due East [3 Marks]
iv) How far is it from O at this instant [2 Marks]
Question Four [20 Marks]
a) Show that
1 2 3
1 2 3
ˆ ˆ ˆ
b b b
a a a
i j k
a × b = given that a a iˆ a ˆj a kˆ
1 2 3 = + + and b b iˆ b ˆj b kˆ
1 2 3 = + +
[4 Marks]
b) Given that a = 4iˆ + 3 ˆj +12kˆ and b = 8iˆ - 6 ˆj find
i. a • b [2 Marks]
ii. The angle between the two vectors a and b [3 Marks]
c) Given that A, B and C are the points (1,1,1), (5,0,0) and (3,2,1) respectively find the
equation which must be satisfied by the coordinates (x, y, z ) of any point P in the
plane ABC. [6 Marks]
d) Find the point of intersection of the lines
4
3
2
2
1
3 +
=
-
=
-
x - y z
and 3x - y + 2z = 8
[5 Marks]
Question Five [20 Marks]
a) Show that
2 2 2 2
A B A B A B
r r r r r r
× + • = [4 Marks]
b) Find an equation for the plane perpendicular to the vector A = 2iˆ + 3 ˆj +16kˆ
r
and
passing through the terminal point of the vector B = iˆ + 5 ˆj +13kˆ
r
. Hence find the
distance from the origin to the plane. [6 marks]
c) Given that A is the point (1,-1,2), B is the point (-1,2,2) and C is the point (4,3,0) ,
[6 marks]
i. find the direction cosine of BA and BC
ii. show that the 69 14 ABC = 0 ¢
d) Find the volume of a parallelepiped whose edges are represented by A = 2iˆ - 3 ˆj + 4kˆ
r
B = iˆ + 2 ˆj - kˆ
r
and C = 2iˆ - ˆj + 2kˆ
r
[4 Marks]
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