Transcript Lecture 2 Vectors and Scalars

L 2 – Vectors and Scalars
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Outline
Physical quantities - vectors and scalars
Addition and subtraction of vector
Resultant vector
Change in a vector quantity, calculating relative
and resolve a vector into components
Vector representation in a component form in a
coordinate system
Vectors in Physics-Examples
Many physical quantities have both magnitude
and direction: they are called vectors.
• Examples: displacement, velocity, acceleration
force, momentum...
Other physical quantities have only
magnitude: they are called scalars.
• Examples: distance, speed, mass, energy...
Displacement and Distance
• Displacement is the vector connecting a
starting point A and some final point B
B
A
• Distance is the length one would travel from
point A to the final point B. Therefore distance
is a scalar
Geometrical Representation of
Vectors
• Arrows on a plane or space
• To indicate a vector we use bold letters
or an arrow on top of a letter
Properties of Vectors
1. The opposite of a vector a is vector - a
2. It has the same length but opposite
direction
3. Two vectors a and b are parallel if one is a
positive multiple of the other:
a = m b,
m>0
Example:
if a = 3 b, then a is parallel to b
(if a = -2 b then a is anti-parallel to b)
Operations: Adding two Vectors
When we add two vectors, we get the resultant
vector a + b, with the parallelogram rule:
Operations: Adding more vectors
• We can add more vectors by pairing them
appropriately
Operations: Vector Subtraction
• Special case of vector addition
 Add the negative of the subtracted
vector
• a – b = a + (– b)
Components of a Vector
• A component is a
part or shadow along
a given direction
• It is useful to use
rectangular
components
– These are the
projections of the
vector along the xand y-axes
Components of a Vector, cont.
• The x-component of a vector is the
projection along the x-axis
 ax = a cosθ
• The y-component of a vector is the
projection along the y-axis
 ay = a sinθ
• a is the magnitude of vector a
 a2 = ax2 + ay2
Example 1
Resolve this
vector along
the x and y
axes to find its
components
respectively.
Example 2
A vector of 15.0 N at 120º to the x-axis is
added to the vector in Example 1. Find the x
and y components of the resultant vector.
15.0 N
The Unit Vectors: i, j, k
A unit vector has a magnitude of 1
i is the unit vector in the x-direction,
j is in the y-direction and
k is in the z-direction.
The Unit Vectors, Magnitude
• Any vector a can be written as:
 a=xi+yj+zk
Example 3 : Given the two displacements
d  6.0i  3.0 j  1.0k
c  4.0i  5.0 j  8.0k
Show that the magnitude of e is approximately
17 units where: e  2.0d  1.0c
Direction of Vectors
N - North
….?
W - West
15 ° east of north, or
75 ° north of east, or
bearing of 15 °
15°
30°
E - East
45°
30°
45 ° west of south, or
45 ° south of west, or
bearing of 225 °
S - South
…..?
Example 4
Find the magnitude and direction of the electric
field vector E with components
3i – 4j.
Note: This vector could
also be written in matrix
form:  3 
 
 4 
Relative velocity
Suppose a cyclist (C) travels in a straight line
relative to the earth (E) with velocity VCE.
A pedestrian (P) is travelling relative to the
earth (E) with velocity VPE.
The relative velocity of the cyclist (C)
with respect to the pedestrian (P) is
given by :

VCP = VCE - VPE
Example 5
A boat is heading due north as it
crosses a wide river with a velocity of
8.0 km/h relative to water. The river
has a uniform velocity of 6.0 km/h due
east. Determine the velocity (i.e. speed
and direction) of the boat relative to an
observer on the riverbank.
The dot (scalar) product
• Imagine two vectors a, b at an angle θ
• The dot product is defined to be:
 a · b = a b cosθ
 Useful in finding work of a force F
CHECK LIST
• READING
Serway’s Essentials of College Physics pages 41-46 and
53-55.
Adams and Allday: 3.3 pages 50-51, 52-53.
Summary
• Be able to give examples of physical quantities
represented by vectors and scalars
• Understand how to add and subtract vectors
• Know what a resultant vector is
• Know how to find the change in a vector quantity,
calculate relative and resolve a vector into components
• Understand how vectors can be represented in
component form in a coordinate system
• Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examples
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Ex 1 – Vx = 8.7N, Vy = 5N
Ex 2 – coordinates of resultant vector are (1.16, 18.0)
Ex 3 – length is 16.9 units, or approximately 17 units
Unknown Directions –
Yellow is 60° S of E, or 30 ° E of S, or bearing 150° (90+60)
Blue is 30° N of W, or 60 ° W of N, or bearing 300° (270+30)
• Ex 4 – Resultant vector E : magnitude 5 units,
– Direction 53° S of E, or 37° E of S, or bearing 143° (90 + 53)
• Ex 5 – velocity of boat relative to earth: magnitude 10 km/hr
– Direction 53° N of E, or 37° E of N, or bearing 37°