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VECTORS
Why Study Vectors?
1. Vectors are the best way to show both
mathematically and graphically the complex
relationsips between appropriate physical
quantities and concepts.
2. Vector representations are concise, explicit
and accurate.
3. Vector rules are uniform and consistent across
vector operations, ensuring correctness of the
description of physical process.
Vectors are mathematical quantities fully
described by both a magnitude and a direction.
1
Because direction is an important characteristic
of a vector, arrows are used to represent them.
The direction that the arrow is pointing
represents the direction of the vector.
The length of the arrow is proportional to the
magnitude of the vector
2,3,4
The tip of the arrow, that is, its point, is called
the head of the vector.
The other end of the arrow is called the tail of
the vector.
5,6
Vectors are usually drawn to scale.
The green arrow is twice as long as the red arrow,
indicating that it has twice the magnitude.
10 m/s
5 m/s
If these were velocity vectors and the green vector
represented a velocity of 10 m/s in the positive x direction,
then the red vector would be interpreted as 5 m/s in the
7
Positive x direction.
Vector Notation
In physics, a vector is usually
named with a single letter with an
arrow above it.
In physics textbooks, the
letter may be simply in a
bold font with no arrow.
The vector has a magnitude of 2 km and a
direction 30° North of East.
If the car in the diagram had moved 4 km
instead of 2 km, the arrow would have been
drawn twice as long.
The practice of using the length of an arrow
to represent the magnitude of a vector
applies to any kind of vector.
Vector Quantities
Displacement
Direction
Velocity
Acceleration
Momentum
Force
Weight
Drag
Lift
Thrust
8
The direction of a vector is often stated in
terms of North, South, East, and West.
Note: The positive or negative on an arrow only indicates its direction,
such as forward (+), backward (-) or upward (+),downward (-). Forces
are neither positive or negative.
For the direction of the arrow to be meaningful,
some sort of coordinate system is necessary
A Cartesian coordinate system is typically used
for this purpose. It consists of a pair of lines on a
flat surface or plane, that intersect at right angles.
9
(x,y)
In the figure above point P1 has coordinates (3, 4), and
point P2 has coordinates (-1, -3).
10
Examples
Note: The positive or negative on an arrow only indicates its
direction, such as forward (+), backward (-) or upward
(+),downward (-). Vectors are neither positive nor negative.
The Physics Classroom
Conventions for Describing Directions of Vectors
We’ve already seen that vectors can be directed
due East, due West, due South, and due North.
But some vectors are directed northeast (at a 45
degree angle); and some vectors are even directed
northeast, yet more north than east.
Thus, there is a clear need for some form of a
convention for identifying the direction of a vector
that is not due East, due West, due South, or due
North.
In cases where the direction of a vector that is not due
East, due West, due South, or due North, the direction of
a vector can be represented as degrees.
11
The Physics Classroom
The direction of a vector can be expressed as a
counterclockwise angle of rotation of the vector about its
“tail” from due East.
Using this convention, a vector with a direction of 40
degrees is a vector that has been rotated 40 degrees in a
counterclockwise direction relative to due east.
A vector with a direction of 240 degrees is a vector
that has been rotated 240 degrees in a
counterclockwise direction relative to due east.
Representing the Magnitude of a Vector
The magnitude of a vector in a scaled vector diagram is
depicted by the length of the arrow. The arrow is drawn a
precise length in accordance with a chosen scale.
For example, the diagram below shows a vector with a
magnitude of 20 miles.
Since the scale used for
constructing the diagram
is 1 cm = 5 miles, the
vector arrow is drawn
with a length of 4 cm.
That is, 4 cm x (5 miles/1
cm) = 20 miles.
Based on the scale, what length would you draw the
vector shown above?
5 cm
Question?
During a relay race, runner A runs a given distance due north
and then hands off the baton to runner B, who runs for the
same distance in a south-easterly direction.
Are they equal?
No, they are not equal!
In order for two vectors to be equal, the magnitude and
direction of both vectors must be the same.
12
Equivalent Vectors
These two vectors are equivalent. They have the
same length and direction.
You can move vectors around on the coordinate
system. So long as you do not change their length
or orientation they are equivalent.
13
A vector having the same magnitude but opposite
direction to a vector A, is -A.
These two vectors are not equal. Even though
their magnitudes appear to be the same, their
directions are not the same.
(a) The displacement vector for a woman climbing 1.2 m
up a ladder is represented by the letter D.
(b) The displacement vector for a woman climbing 1.2 m
down a ladder is represented by the letter -D
(a) The force vector for a man pushing on a car with 450
N of force in a direction due east is F.
(b) The force vector for a man pushing a car with 450
N of force in a direction due west is –F.
Note: The positive or negative on an arrow only indicates its direction,
such as forward (+), backward (-) or upward (+),downward (-).
Forces are neither positive or negative.
P and Q are two vectors represented in the
diagram below.
P and Q are represented as two vectors having
equal magnitudes but opposite directions
therefore, |P| = |Q|.
Addition of Collinear Vectors
Points are said to be collinear if they lie on a
single straight line, especially if it is related
to a geometric figure such as a triangle.
We are thus going to learn to add vectors that are
in a straight line.
14
The order in which you add vectors does
not matter.
Addition of vectors:
Two or more vectors may be added together to
produce their sum.
If two vectors have the same direction, their
resultant has a magnitude equal to the sum of their
magnitudes and will also have the same direction.
15, 16
(a) The man is pushing on the desk with a force of 40 N
to the right. The boy is pushing on the desk in the
same direction with a force of 20 N.
(b) The total force the man and the boy are applying to
the desk to the right is shown by the vector b. Since
both force vectors are in the same direction, they
are added
Subtraction of vectors:
Two or more vectors may be subtracted to
produce their difference.
If two vectors have opposite direction, their
resultant has a magnitude equal to the difference
of their magnitudes. The direction will be
determined by the magnitude of the largest
vector.
17, 18
(a) The man is pushing on the desk with a force of 40
N to the right. The boy is pushing on the desk in
the same direction with a force of 20 N.
(b)The girl is pushing in the opposite direction with a
force of 10 N
(c) The difference of the vectors is 50 N.
I walk 100 m north and then 200 m south. What
is my total displacement from my starting point?
100 m N – 200 m S =
-100 m South
In the example to the left,
the two vectors are parallel.
How would you add or subtract
vectors that are perpendicular?
Triangle law of vector addition
In the diagram below, two perpendicular vectors are
represented by two sides of a triangle in sequence.
In sequence means that the vectors are placed such that
the tail of vector BC begins at the arrow head of the
vector placed before it, AB.
The third closing side of the triangle (AC), represents the
sum (or resultant ) of the two vectors in both magnitude
and direction.
In this case,the closing
side of the right triangle
AC represents the sum (i.e.
resultant) of individual
displacements AB and BC.
AC = AB + BC
Draw the resultant vector
by connecting the tail of
the first vector to the
head of the last vector..
The direction of the resultant vector must be included. The
vector’s direction is show as an angle, measured from the
horizontal.
If drawn to scale, a ruler is used to measure the magnitude of
the resultant vector and a protractor is used to measure the
angle  (direction) of the resultant vector.
The triangle law does not restrict with which vector or where
to start.
Also, it does not put conditions with regard to any specific
direction for the sequence of vectors, like clockwise or
counter clockwise, to be maintained.
In figure (i), the law is applied starting with vector,b.
In figure (ii) the law is applied starting with vector,a.
In either case, the
resultant vector, c, is
same in magnitude
and direction.
Tail Wind
The Physics Classroom
Head Wind
Cross Wind
These vectors are not
parallel, they are
perpendicular.
If two vectors are
at right angles to each
other, to get their sum,
you need to find
the resultant.
19
The resultant vector is the vector that “results” from
combining two or more vectors together.
The resultant is the diagonal of a rectangle constructed
with the two vectors as sides.
20, 21
Step technique to find the resultant of a pair of vectors
that are at right angles to each other.
Draw the two
vectors with
their tails
touching.
Draw a parallel
projection of
each vector
with dashed
lines to form a
rectangle.
Draw the diagonal
from the point
where the two
tails are touching.
22
1. Draw the two
vectors with
their tails
touching.
2. Draw a parallel
projection of
each vector
with dashed
lines to form a
rectangle.
3. Draw the diagonal
from the point
where the two
tails are touching.
Which vector shown below is the resultant?
b is the resultant. Components will have
their tails touching.
The equation is a + c = b
Now that we have drawn the resultant, how do
we figure out the new speed?
If two vectors are
at right angles to
each other, we can
use the Pythagorean
Theorem
90°
(a2 + b2 = c2) to
solve for the
resultant
.
23
a2 + b2 = c2
Resultant2 = (60 km/h)2 + (80 km/h)2 =
= 3600 (km/h)2 + 6400 (km/h)2
=
= 100 km.h
(km/h)2
I walked 40 km east then 30 km north. What's my final
displacement?
We first walk 40 km
east. Draw the vector.
From the tip of the old
arrow, draw the 30
km north.
To find the final
displacement, draw an
arrow from the end of
the first vector to the
tip of the last vector.
d
30
40
Using the Pythagorean
Theorem, our total
displacement is 50 km NE.
(40)2 + (30)2 = d2
1600 + 900 = d2
2500 = d2
= d
50 km North of East = d
Let’s Practice
R2 = (5)2 + (10)2
R2 = 125
R = SQRT (125)
R = 11.2 km
R2 = (30)2 + (40)2
R2 = 2500
R = SQRT (2500)
R = 50 km
Special Case
For any square, the length of the diagonal is the square
route of two, (
) or 1.414,times either of the sides.
If the rectangle
formed is a square, we
use the
as the
resultant.
24
For any square, the length of the diagonal is
or 1.414, times either of the sides.
,
1.414 x 100 = 141.4
Let’s Practice
1.414 x 75 = 106.05
A plane can travel with a speed of 80 mi/hr with
respect to the air. Determine the resultant
velocity of the plane (magnitude only) if it
encounters a
1. 10 mi/hr headwind
70 mi/hr
2. 10 mi/hr tailwind
90 mi/hr
3. 10 mi/hr crosswind
80.6 mi/hr
4. 60 mi/hr crosswind 100 mi/hr
1. A hiker walked walked 3.47 km [E] and 5.32
km [N]. What distance (in km) did the hiker
walk?
8.79 km
2. What was the magnitude (in km) of the
hiker's displacement in the previous question?
6.35 km
Vector representation review:
1. The reference direction is indicated.
2. Vectors are drawn to scale and the scale is indicated.
3. The vectors are represented as arrows with a length
proportional to their magnitude and are correctly
orientated with respect to the reference direction.
4. The direction of the vector is indicated by an
arrowhead.
5. The arrows should be labeled to show which vectors
they represent.
Part 2 Vector
Components
Any vector directed in two dimensions can be
thought of as having two parts.
Each part of a two-dimensional vector is known as
a component.
In this case, two dimensional means that it has both an upward and a
Rightward component.
25
The process of determining the components of
a vector is called resolution.
Any vector drawn on a piece of paper can be
resolved into vertical and horizontal components.
26
The Physics Classroom
If Fido's dog chain is stretched upward and rightward and
pulled tight by his master, then the tension force in the
chain has two components - an upward component and a
rightward component.
The Physics Classroom
The upward and rightward force on the chain is equivalent
to an upward force and a rightward force by two chains.
The Physics Classroom
Steps to resolve a vector into components!
V
V
• Vertical lines are drawn
from the tail of the
vector (top)
2. A rectangle is drawn
that encloses the vector
V as its diagonal.
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Component x
Music:
Linkin Park - Hit The Floor
Metallica
- Fuel
Jeopardy Theme
Sources:
The Physics Classroom
Conceptual Physics – Paul Hewitt
Physics – Cutnell and Johnson, third edition
The term “sequence” means that the vectors are
placed such that the tail of a vector begins at the
arrow head of the vector placed before it.