Introduction to Vectors

1
Introduction to Vectors

• COORDINATE SYSTEM AND FRAMES OF REFERENCE

A coordinate system consists of
1. A fix origin, O.
2. A set of specified axes or directions with
appropriate scales labels.
O
3. Instructions on labelling a point relative to
the origin and the axes.
2
Cartesian Coordinate System
Axes at right angles to each other for the basis
of this system
P (x,y)
y
O
x
Up to three axes can be used.
The relative positive direction can be determined
by the right hand rule.
3
Polar Coordinate System
The distance (r) from the origin to the point is
measured and the angle between this line and a
fixed axis is measured.
P (r,q)
r
O
Up to three axes can be used.
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Relationship between the two systems

• Is determined by basic trigonometry.

x r cosq
y r sinq
r
r
y
Tanq y/x
q
q
O
r2 x2 y2
x
x
5
Excercise

• Two points in a plane have polar coordinates
  (2.5m, 300) and (3.5m, 1200). Find their
  Cartesian coordinates?

For point 1.
For point 2.
x1 r1 cosq1
x2 r2 cosq2

• 2.5.cos(30)

• 3.5.cos(120)

• 2.165m

• -1.75m

y2 r2 sinq2
y2 r2 sinq2

• 2.5.sin(30)

• 3.5.sin(120)

• 1.25m

• 3.03m

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Excercise

• Two points in a plane have polar coordinates
  (2.5m, 300) and (3.5m, 1200). What is the
  distance between them?

Answer 4.3 m
R13.5m
q21200
R12.5m
q1300
O
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Vectors and Scalars

• A Scalar quantity is one which can be fully
  described by its magnitude only.
• E.g. temperature, mass, volume, time and speed.
• A vector quantity is one which require both the
  magnitude and direction for its full description.
• E.g. velocity, displacement, acceleration and
  force.

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Properties of Vectors

• Two vectors are equal if they have the same
  magnitude and direction.

All of the vectors in the above diagram are equal
even though they have different starting points.
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Adding Vectors

• To add two or more vectors of the same quantity
  do the following.

1. Draw the first vector
2. Then draw the second vector from the tip of
the previous vector.
3. Repeat this process until all vectors are
drawn.
4. The resultant vector is the line from tail of
the first vector to the tip of the last vector
5. The resultant vector can be determined by the
following methods- 1. Triangulation 2.
Components
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Adding Vectors - graphically
Example Draw the resultant vector for vrv1v2v3
v1
v3
vr
v2
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Adding Vectors - Triangulation
The resultant vector for adding two vectors is
given by- OB2OA2AB2-2OA.AB.cos(AOB)
Y
B
A
0
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Adding Vectors - Component

• Break each vector into its components in
  indirections which are at right angles to each
  other. Add all the components in each direction
  together then combine the components back into to
  a single vector.

Y
B
A
0
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Subtracting Vectors

• To subtract a vector from one another, add the
  negative of the vector to be subtracted.

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Adding Vectors

• Example A car travels 20 km due north and then
  35 km at 600 west of north. Calculate its final
  displacement from its starting point.

35 km
25 km
Answer 48 km at 39o to the west of north
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Relative Velocity

• The displacement, velocity and acceleration
  measured by the observer depends on the frame of
  reference of that observer.

That is two observers moving with respect to each
other will generally report different
measurements of these quantities for the same
event.
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Relative Velocity

• Example A person riding in a car throws the ball
  up in the air while the car is moving. Another
  person standing beside the car also watches the
  ball.

Person standing sees the ball moving in a
parabola.
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Relative Velocity

• Example A person riding in a car throws the ball
  up in the air while the car is moving. Another
  person standing beside the car also watches the
  ball.

Person in the car sees the ball going straight
up, then down.
18
Relative Velocity

• Exercise Rain is falling steadily on a wind free
  day in Melbourne with a vertical speed of
  approximately 25 km/hr. A VU student is driving
  her car at 60 km/hr on the way to the University.
  How does the driver see the rain?

What do we know?
Velocity of rain wrt driver Vrd?
What do we need to know?
Answer The driver sees the rain hitting the
windscreen at 65 km/hr at 67.40 to the
vertical.