VECTORS
BISHOPS
Grade 10
Physical Science
VECTORS AND SCALARS
A scalar is a physical quantity that has magnitude only. Examples: time, distance, speed, mass, energy, work and power.
A vector is a physical quantity that has both magnitude and direction. Examples: displacement, velocity, acceleration,
weight and momentum.
Illustration of the difference between a vector and a scalar:
A girl walks a distance of 800 m along a winding path from school to home. The straightline distance of her home from the school is 500 m and the direction is north-east. Write
down the values for the distance and displacement of her home from the school.
Distance = 800 m
(Magnitude only)
Displacement = 500 m, NE
(Magnitude and direction)
Home
800 m
N
500 m
School
Distance is the magnitude of the actual distance covered by a moving object from the
starting point to the finishing point.
Displacement is the magnitude and direction of the straight-line distance from the starting point to the end point of a
motion.
REPRESENTATION OF DIRECTION
There are two methods of representing direction:
1.
The use of the points of the compass as a reference from which to measure angles:
Example: 30o east of north or N 30o E.
2.
The use of bearings, i.e. the clockwise measurement of angles from north or the vertical.
Example:
Bearing 210o
N
210o
REPRESENTATION OF VECTORS
Graphical representation:
A vector is represented graphically by an arrow drawn to scale. The magnitude of the vector is represented by the length of
the arrow and the direction of the vector by the direction of the arrow.
When drawing a vector, the following must be shown:
(1)
the reference direction, e.g. north;
(2)
the scale used;
(3)
an arrow head to show direction.
Symbolic representation:
The symbol of a vector is distinguished from the symbol of a scalar by means of an arrow written above the symbol of a
vector.
Example:
Distance (scalar):
Displacement (vector):
s

s
Speed (scalar):
Velocity (vector):
v

v
ADDITION OF VECTORS
Vectors can be added by algebraic addition if they act along the same straight line or by accurate geometric construction or
trigonometric calculation if they operate at an angle to each other. The answer to a vector addition is called the resultant
which is the vector from the tail of the first vector to the head of the last vector.
A resultant of a number of vectors is the single vector that will have the same effect as all the original vectors acting
together.
1.
Addition of vectors acting along the same straight line by algebraic summation
A vector is considered negative with respect to another if it acts in the opposite direction. One therefore has to state
which direction is positive.
Example:
Two vectors of 12 N and 5 N act in opposite directions along the same straight line. Calculate the resultant.
By calculation:
Let the direction of the 12 N vector be positive.
Resultant = +12 N – 5 N = + 7 N, i.e. 7 N in the direction of the 12 N vector.
Resultant
7N
5N
12 N
1
2
3
4
5
6
7
8
9
10
11
2.
Addition of two vectors at an angle to each other (For extension if time allows)
2.1
By geometric construction
12
The following two methods can be used:
2.1.1 In the triangular method, the vectors are constructed head to tail and the third side of the triangle is completed by
drawing an arrow from the starting point to the finishing point. The magnitude and direction of the third side as
taken from the starting point is the resultant.
Example: A girl walks a distance of 100 m due east before turning and walking a distance of 50 m in a direction E
60o S. Determine her resultant displacement by accurate construction and measurement.
Scale: 1 cm = 10 m
Bearing = 90o + 60o = 150o
100 m
150o
o
19,1
Res u
lt
an t
50 m
Resultant displacement = 132,3 m, E 19,1o S or 132,3 m, 109,1o
(The above is a computer generated sketch diagram and is therefore not an accurate construction.)
2.1.2
In the parallelogram method, the vectors are constructed tail to tail, the parallelogram is completed and the
resultant is determined by measuring the magnitude and direction of the diagonal from the starting point.
Scale: 1 cm = 10 m
Bearing = 90o + 60o = 150o
100 m
o
o
150 19,1

Res u
lt

an t

50 m

Resultant displacement = 132,3 m, E 19,1o S or 132,3 m, 109,1o
2.
Addition of more than two vectors by accurate geometric construction
In the polygon method, vectors are constructed head to tail and the length and direction of the line drawn from the
starting point to the finishing point represents the resultant.
Example: Three vectors of 3 N, 60o, 4 N, 90o and 5 N, 230o act at a point. Determine the resultant by accurate
construction and measurement.
Scale: 1 cm = 1 N
4N
230o
60o
3N
5N
R
Resultant = 3,3 N, 121o
Note: The larger the scale, the smaller the margin of error.
It is best to draw a rough sketch before starting your construction in order to determine the position of the
starting point so that the construction does not run off the page.
EXERCISE
1.
Which of the following are both vector quantities?
A
time and distance
C
velocity and momentum
B
speed and energy
D
momentum and time
2.
Which of the following quantities is not a vector?
A
force
C
velocity
B
displacement
D
energy
3.
The resultant of two forces acting on a body
A
acts at the same point but in the opposite direction.
B
keeps the body in stationary.
C
produces the same effect as the two forces together.
D
has the opposite effect of the two forces together.
4.
Two cyclists travelled from town X to town Y. Their average velocities between the two towns were the same, but
their average speeds differed. This is possible if
A
they travelled on different roads.
B
they travelled at different times.
C
their distances and displacements were equal in magnitude.
D
they travelled on the same road, but took different times to travel from X to Y.
5.
If a body is displaced 24 m horizontally and then 32 m vertically, the magnitude of the resultant displacement, in
metres, is
A
1,3.
B
8.
C
40.
D
56.
6.
In which of the following vector diagrams is the resultant of the three vectors zero?
A
B
C
D
7.
The accompanying diagram represents three forces x, y and z. Which one of the following vector equations
describes the situation?
A

x+
y=
z
B
  
x=y+z
C
  
x+y+z=0
D
  
y=x+z
x
y
z
8.
Two forces X and Y are acting in opposite directions. X and Y can be replaced by a single force of magnitude 5 N.
If the magnitude of X is 2 N, which one of the following can be the magnitude of force Y?
A
1N
B
3N
C
7N
D
10 N
9.
A ship is sailing slowly due west. In which direction
should a passenger walk on the deck at P, at a suitable
speed, to move due south?
A
along PD
B
along PB
C
along PC
D
along PA
A
C
West
P
South
B
D
10.
If three vectors acting along the same line with magnitudes 30, 40 and 50 are added together, which one of the
following cannot represent the magnitude of the resultant?
A
30 N
B
60 N
C
20 N
D
120 N
16.
The vector that best represents the resultant of the forces F1 and F2 shown acting on point
P in the diagram is
A
B
C
F1
D
P
17.
A force is represented by AB in the diagram. This force (AB) is the resultant of a force
represented by AD and a force F which is not shown. Force F is direction from
A
D to B.
B
B to D.
C
A to C.
D
C to A.
D
F2
C
A
18.
A man travels 50 km due west, then 40 km due south and finally 20 km due east. The magnitude of his
displacement, in km, is
A
30.
B
50.
C
60.
D
90.
19.
A person walks 5 km north-east and then 5 km south-east. Her displacement is
A
10 km.
B
7,07 km.
C
10 km east.
D
7,07 km east.
B