Scalars & Vectors
Name: ________________
Class: _________________
Index: ________________
Learning objectives
At the end of this unit you should be able to :
1. State what is meant by scalars and vectors
quantities and give common examples of each;
2. Add two vectors to determine a resultant (graphical
method);
3. Solve problem for a static point mass under the
action of 3 forces for 2 dimensional cases (graphical
method).
Scalar quantities are quantities that have magnitude
only.
Vector quantities are quantities that have both
magnitude and direction.
Scalars
Vectors
Distance
Displacement
Speed
Velocity
Mass
Weight
Time
Acceleration
Pressure
Force
Energy
Momentum
Volume
Density
Addition of Scalar Quantities
The addition of scalar quantities is very simple as
it involves addition or subtraction using simple
arithmetic. An example is 4 kg plus 6 kg always
gives the answer 10 kg.
Addition of Vector Quantities
The addition of vector quantities is not as straight
forward as that of scalar quantities as the direction
of vector quantities must be considered.
Adding parallel vectors
4N
6N
Resultant force on block = 4 + 6 = 10 N to the
right
4N
6N
Resultant force on block = 6 - 4 = 2 N to the left
Observations: Vectors can be represented graphically by
arrows. The length of the arrow represents the magnitude of
the vector. The direction of the arrow represents the
direction of the vector.
Vector Addition (Method 1 – Tip to tail)
• The resultant (force) of two vectors is a single vector which
produces the same effect as given two vectors.
• To add vectors P and Q, the starting point of vector P is
placed at the ending point of vector Q. The vector sum or
the resultant vector of P and Q is represented by vector R,
which is the vector joining the starting point of vector P to
the ending point of vector Q. The same resultant vector is
obtained irrespective of whichever order the vectors are
P
added.
Q
Q
R
P
Vector Addition (Method 2 - Parallelogram)
• Vectors acting at an angle to each other can be added
graphically using the parallelogram law.
• The parallelogram law of vector addition states that if two
vectors acting at a point are represented by the sides of a
parallelogram drawn from that point, their resultant is
represented by the diagonal which passes through that point
of the parallelogram.
P
P
Q
R
Q
Example
A force of 3 N acts at 90o to a force of 4 N. find the
magnitude and direction of their resultant, R.
Solution
R2 = 32 + 42 = 25
R = 5N
tan  = 3/4
 = 37o
Scale : 1 unit rep. 1 N
3N
R


4N
Example
A force of 10 N acts at 45o to a force of 4 N. find the
magnitude and direction of their resultant, R.
Solution
Scale : 1 unit rep. 2 N
From the diagram, R = 6.2 units = 12.4 N
( =13° counter-clockwise from 10 N force)
4N
R
450

10 N
Application
Figure A below shows a catapult used to project an object. Force F
pulls back the object, creating tension in the rubber cords. The
tension force in each rubber cord is 20N and the two cords are at
60° to each other. Fig. B shows the direction of the two tension
forces acting on the object. By making a scale drawing of Fig. B,
find the resultant of these two tension forces acting on the object.
State the scale of that you use.
Solution (Parallelogram Method):
Scale: Let 1.0 cm rep. 4 N
8.6 cm
5 cm
60°
5 cm
Since 1cm rep 4 N, the resultant
force is 8.6 x 4 = 34.4 N
Application
Figure below shows the top view of the wooden block. Two forces,
magnitude of 30 N and 20 N, act on the wooden block (not drawn to
scale). By means of a scale diagram, determine the magnitude and
direction of the resultant force acting on the wooden block.
30 N
40
15
20 N
Solution (Tip to tail Method):
Scale: Let 1cm rep. 10 N
55°
2 cm
3 cm
4.5 cm
Since 1cm rep 10 N, the resultant
force is 4.5 x 10 = 45 N at an
angle of 25° clockwise from the
30N force
Tip to tail method
References
http://www.jaunted.com/tag/Singapore%20Flyer