Ms. Arra C. Quitaneg
What do we measure to
tell how hot or how cold
an object is?
What describes how
much space is occupied
by an object?
What is used to
describe how
massive an object is?
Temperature
Volume
Mass
A physical quantity which is described by a
single number with a unit
Examples: time, temperature,mass, density
electric charge
Calculations with scalar quantities use
operations of ordinary arithmetic .
A physical quantity which has both
magnitude and direction.
When handwritten, use an arrow:
When printed, will be in bold print with an
arrow:
A
A
When dealing with just the magnitude of a
vector in print, an italic letter will be used:
A or | |
The magnitude of the vector has physical
units
The magnitude of a vector is always a
positive number
A
Displacement
Force
Velocity
Acceleration
A vector is represented by an arrow
Arrowhead points to the direction.
Length of the arrow represents the
magnitude of the vector.
Vector’s magnitude can also be expressed in
numbers.
The magnitude of a vector is always positive.
Direction of the vector can be represented
by a sign ( + or - ) depending on the sign
convention ex: North +, South –
Direction is also expressed in words.
Ex. South, Southwest
Angle is also used to describe a vector’s
direction.
ex. 300 South of East, + 500 , + 1200 , - 2200
Scale: 1 cm = 1 m
d= 4m , East
Scale: 1 cm = 1 m
d= 4m , 45 0 North of East
A = 4 m, 50 0 N of E
A =4 m, 400 E of N
Parallel vectors- if two vectors have the same
direction
Equal vectors- if two vectors have the same
magnitude and direction
Negative of a vector- a vector having the same
magnitude as the original but has opposite
directions.
Antiparallel – when two vectors have opposite
directions, whether their magnitudes are
equal or not.
Scale: I cm= 1 unit
4 Newtons East
5 m, North
10 m/s, 20 0 N of E
8 m, 40 0 E of N
7 N, 40 0 W of S
6 m, 50 0 S of E
12 m, 40 0 S of W
5m, + 230 0
8 m/s, - 1800
Result of vector addition is called as vector
sum or resultant vector.
R = (A +B) + C = A + (B +C)
It obeys the associative law, order of addition
makes no difference.
R = A + B = B + A
It obeys commutative law, order of terms
in a vector sum does not matter.
Head to tail method
1. From the origin, the vector, based
on its direction and magnitude is
drawn.
2. Then, the tail of the second
vector is attached to the head of
the first.
3. Resultant vector is drawn from the
tail of the first vector to the head
of the last vector.
When vectors are parallel, just add magnitudes and keep the direction.
Ex: 50 m/s east + 40 m/s east = 90 m/s east
When vectors are antiparallel, just subtract the smaller magnitude from the larger and use the direction of the larger.
Ex: 50 m/s east + 40 m/s west = 10 mph east
R
θ
Use a protractor to measure the angle.
A = 5 N, East
B = 7 N, 60 deg N of E
Find the resultant
of A and B. A = 11 N @ 35° N of E
A 35° N of E
B = 18 N @ 20° N of W
B
20° N of W
R
57° N of W
R = 14.8 N
@ 57° NW
Parallelogram method
1. Plot the given vectors from the origin
( based on their magnitude and directions
2. Reflection of the vectors are drawn, until a
parallelogram is formed.
3. Resultant vector is the diagonal of the
parallelogram. Measure the length and the
angle of the resultant vector.
VECTOR ADDITION EXERCISES:
1.Erwin walks 300 East and stops to rest
and then continues 400 m East. What is
his total displacement?
2.Mimi walks home from school 300 m East
and remembers that she has to bring
home her Science book which a classmate
borrowed. She walks back 500m West to
her classmate’s house. What is the
resultant displacement of Mimi?
3. Early in the morning, Patrick would
always jog 500 m East from his house
and turns North and walks 300 m. What
is his displacement from his house?
4. An ant crawls on a tabletop. It moves 2
cm East and turns 3 cm 40 0 North of
East and finally moves 2.5 cm North.
What is the ant’s total displacement?
The analytical
method of vector
addition!!!
Magnitude of the resultant vector is given by the arithmetic sum of the magnitudes of the individual vectors.
Direction is unchanged.
Magnitude of the resultant vector is given
by the arithmetic difference of the
magnitudes of individual vectors.
Direction of the resultant vector is the
same as the direction of a vector with
greater magnitude.
Use Pythagorean theorem
22
yx AA A
The Pythagorean theorem
Trigonometric functions are used in
solving for the directions.
sin = opp/ hyp
cosine = adjacent/ hyp
tan = opp/ adjacent
SOH CAH TOA
x
y
A
A1tan
Component method-
1. Draw each vector.
2. Find the x and y components of
each vector ( Vector resolution).
Use trigonometric functions.
3. Find the sum of x – components
and y components.
4. Use pythagorean theorem to get
the resultant vector.
The river was moving with a velocity of 3
m/s, North and the motor boat was
moving with a velocity of 4 m/s, East.
What would be the resultant velocity of
the motor boat (i.e., the velocity relative
to an observer on the shore)?
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
a. 10 mi/hr headwind.
b. 10 mi/hr tailwind.
c. 10 mi/hr crosswind.
d. 60 mi/hr crosswind.
A motor boat traveling 5 m/s, East
encounters a current traveling 2.5 m/s,
North.What is the resultant velocity of the
motor boat?
What are the x and y components a force of
500 N exerted at an angle 300 from the
horizontal.
A man walked 5m, 400 North of East. What
are the x and y components of his
displacement?
A man walked 2m, 250 North of East, then 5m
300 West of North. What is the man’s
displacement? (magnitude and direction)
An airplane with a velocity of 120
km/h,heads North. However wind blows with
a speed of 30 km/h at an angle of 400 North
of West. What is the velocity of the airplane
relative to the ground?
An airplane has a speed of 285 km/h with respect to
the air. There is a wind blowing at 95 km/h at 30 deg
north of east with respect to Earth. In which
direction should the plane head in order to land at
an airport due north of its present location. What
would be the plane’s speed with respect to the
ground?
You are piloting a small plane and you want to reach
an airport 450 km due south in 3 hours. A wind is
blowing from the west at 50 km/h. What heading
and airspeed should you choose to reach your
destination in time?
An airplane,moving at 375 m/s relative to the
ground, fires a cannon shell at a speed of 782 m/s
relative to the plane. What is the speed of the shell
relative to the ground.
1. Solve for the x and y components of each given
vector. Indicate the sign of each component.
2. Add all x components, add all y components.
3. Use pythagorean theorem to get the magnitude
of the resultant vector.
4. Solve for the direction using tan theta function.
5. Express your answers in 2 decimal places.