a) Which pair(s) of vectors form opposite vectors? _______________________

1
MTH-5110 Pretest Solutions
1. Five vectors are represented on the
following Cartesian plane.
a) Which pair(s) of vectors form opposite
vectors? _______________________ b) Which
pair(s) of vectors form orthogonal vectors?
______________________ c) Name the vector(s) that
could form a vector basis with vector w.
___________
2. From the three expressions given below,
determine which one is true. State the property
associated with the true expression.
C is true. It demonstrates the distributive
property of multiplication over addition of
vectors.
2

1. A small plane flies in a direction N 35 W at 250
   km/h. A wind blows to the south at 55 km/h to
   change its direction. Construct a vector diagram
   to show the resultant.

50 km/h

1. Given the following ordered pairs A (-1, 3), B
   (3.5, 4), C (5, -2) and D (1, -5), use the
   algebraic method to calculate the components of
   the resultant vector, r.

3

1. Two forces, one 150 N and the other 220 N are
   applied to the same object. The first force is
   oriented SW and the second at 125. What single
   force would have the same effect on this object?

The resultant components indicates that the
vector is in quadrant II. The angle of
orientation can be determined using the following
formula
A force of 243.8 N in the direction of 134º would
have the same effect as the two forces described
above.
4

1. A swimmer crosses a river in a direction S 30 W
   at a speed of 2.5 km/h. A current of 1 km/h
   towards the east changes the swimmers direction.
   What is the angle of deviation from the intended
   direction.

Components for swimmer
Because the direction of the current is east,
there is no vertical component. The 1 km/h is
only a horizontal component.
The resultant components indicates that the
vector is in quadrant III. The angle of
orientation can be determined using the following
formula
Instead of swimming S 30º W, the current causes
the swimmer to swim S 7º W.
5

1. Calculate the components of the resultant vector
   w.

1. Determine coefficients k1 and k2 of this linear
   combination.

6

1. Solve the following problems.

1. A mother pulls her child in a sleigh. The rope
   on the sleigh makes a 43 angle with the
   horizontal and she pulls with a force of 190 N to
   cause it to move. How much work (in kilojoules)
   is done to make the sleigh move 150 m.

S 35º E is 270º 35º 305º The angle between
vectors u and v is 305º 170º 135º
7

• Using vector theory, complete the following
  proof.
• In a right triangle, the measure of a side
  adjacent to the right angle is the geometric mean
  of its projection and the hypotenuse.

Approach We can apply the definition of a norm
of a vector to side AB and rewrite side AB as the
sum of vectors using Chasles relation. The
expression can be transformed to eventually reach
the sired conclusion.
Chasles Relation
Distributive Property of multiplication over
addition
By Hypothesis 1
Chasles Relation
Distributive Property of multiplication over
addition
By Hypothesis 2
Scalar Product of collinear vectors
8

1. Prove that quadrilateral ABCD represented below
   is a rhombus.

Quadrilateral ABCD is a rhombus.

1. Using the vectors below, prove that the addition
   of vectors is associative.

9

1. Triangle ABC is a right triangle whose altitude
   is segment AD. Determine whether the statements
   concerning this figure are true or false. Adjust
   the false statements to make them true.

False
True
False
True
False

1. A ferry is used to transport passenger vehicles
   between Cap-Rouge and Cap-Espoir, a distance of 8
   km. Its direction with respect to Cap-Rouge is N
   35º E. The current of the river that the ferry
   crosses is 3.5 km/h to the west. So as to land
   at the Cap-Espoir in one hour, at what speed and
   in what direction should the captain set out in
   the ferry?

CAP-ESPOIR
N
We know the resultant and are asked to determine
the original vector for the ferry.
Direction of current
8 km
35º
E
W
CAP-ROUGE
S
Because the direction of the current is west,
there is no vertical component. The 3.5 km/h is
only a horizontal component but in a negative
direction.
The ferry should head out at a speed of 10.41
km/h at an angle of 39º.
10

1. Given the following ordered pairs A (-5.2, -3),
   B (-3, 2.5), C (-1, 7) and D (3.3, -4), Using the
   scalar product, calculate the angle between
   vectors s and t.

The angle between the two vectors is 43º.