Physics: Problem Solving
Chapter 4
Vectors
Physics: Problem Solving
Chapter 4
Vectors
Chapter 4: Vectors
Vector Review
Trigonometry for Physics
Vector Addition—Algebraic
Vector Resolution
Chapter 4: Vectors
Vector:
A measurement with both magnitude
and direction
Magnitude:
A numerical value
Direction:
+/–
North, South, East, West
Which is larger 2m/s or – 3 m/s?
Chapter 4: Vectors
Vector: What does it mean when an
object has…..?
Velocity—negative and positive
Acceleration—negative and positive
Chapter 4: Vectors
Vectors can be added together both
graphically and algebraically
Graphic addition: using arrows of
proper length and direction to add
vectors together
Algebraic addition: using
trigonometry to add vectors together
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
1. Draw first vector to scale (magnitude &
direction)
2. Draw second vector to scale. Connect the
“butt” of the second vector to the “head”
of the first vector
3. Repeat Step #2 until you run out of
vectors.
4. Draw Resultant (?). Connect the “butt” of
the first vector with the “head” of the last
vector—butt-butt, head-head
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
A dog walks 15 km east and then 10 km
west find the sum of the vectors
(resultant)
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
A dog walks 15 km east and then 10 km west
find the sum of the vectors (resultant)
1. Draw first vector to scale (magnitude &
direction)
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
A dog walks 15 km east and then 10 km west
find the sum of the vectors (resultant)
2. Draw second vector to scale. Connect the
“butt” of the second vector to the “head”
of the first vector
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
A dog walks 15 km east and then 10 km west
find the sum of the vectors (resultant)
4. Draw Resultant (?). Connect the “butt” of
the first vector with the “head” of the last
vector—butt-butt, head-head
Measure: 5 km east
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
A dog walks 15 km east and then 10 km
south find the sum of the vectors
(resultant)
Measure: 18 km
34 south of east
Chapter 4: Vectors
Graphic addition:
“Butt-Head” Method
Examples
A shopper walks from the door of the
mall to her car 250 m down a row of
cars, then turns 90 to the right and
walks another 60 m. What is the
magnitude and direction of her
displacement from the door?
Answer
257 meters 13.5 from door
Chapter 4: Vectors
This presentation deals with adding
vectors algebraically
But first…..you need to learn some
trigonometry!!!
Vector Addition-Graphic
Let’s test our knowledge! (1-5)
All the Trig. you need to know
for Physics (almost)
This is a right triangle
All the Trig. you need to know
for Physics (almost)
This is an angle (-theta) in a right
triangle

All the Trig. you need to know
for Physics (almost)
This is the hypotenuse (H) of a right
triangle

All the Trig. you need to know
for Physics (almost)
This is the side adjacent (A) to the
angle (-theta) in a right triangle

All the Trig. you need to know
for Physics (almost)
This is the side opposite (O) the
angle (-theta) in a right triangle

All the Trig. you need to know
for Physics (almost)
Summary
Opposite
(O) side
Angle 
Adjacent (A) side
All the Trig. you need to know
for Physics (almost)
SOH, CAH, TOA Trigonometry
Used when working with right triangles
only!
Opposite
(O) side
Angle 
Adjacent (A) side
All the Trig. you need to know
for Physics (almost)
SOH
sin  = O/H
Opposite
(O) side
Angle 
All the Trig. you need to know
for Physics (almost)
Example:
Find the angle of a right triangle
which has a hypotenuse of 12m and a
side opposite the angle of 9m.
Opposite (O)
side = 9m
Angle  = ?
All the Trig. you need to know
for Physics (almost)
Example:
sin  = O/H
sin  = 9/12 = 0.75
 = sin–1 0.75 = 48.6°
Opposite (O)
side = 9m
Angle  = ?
All the Trig. you need to know
for Physics (almost)
CAH
cos  = A/H
Angle 
Adjacent (A) side
All the Trig. you need to know
for Physics (almost)
Example:
Find the hypotenuse of a right
triangle which has an angle of 35°
and a side adjacent the angle of
7.5 m.
Angle  = 35°
Adjacent (A) side = 7.5m
All the Trig. you need to know
for Physics (almost)
Example:
cos  = A/H
cos 35 = 7.5/H
H = 7.5/cos 35° = 7.5/0.819 =
9.16m
Angle  = 35°
Adjacent (A) side = 7.5m
All the Trig. you need to know
for Physics (almost)
TOA
tan  = O/A
Opposite
(O) side
Angle 
Adjacent (A) side
All the Trig. you need to know
for Physics (almost)
Example:
Find the side opposite the 35° angle
of a right triangle which has a side
adjacent the angle of 7.5 m.
Opposite
(O) side
Angle  = 35°
Adjacent (A) side = 7.5m
All the Trig. you need to know
for Physics (almost)
tan  = O/A
tan 35 = O/7.5
O = (7.5)(tan 35) = (7.5)(.7)=
5.25m
Opposite
(O) side
Angle  = 35°
Adjacent (A) side = 7.5m
All the Trig. you need to know
for Physics (almost)
SOH, CAH, TOA Trigonometry
Remember the Pythagorean Theorem
Work on Trig. Worksheet—due
tomorrow
Opposite
(O) side
Angle 
Adjacent (A) side
Vector Addition—Algebraic
When adding vectors together
algebraically using trigonometry it is
important to use the Physics Problem
Solving Technique
Vector Addition—Algebraic
Example:
A car moves east 45 km turns and
travels west 30 km. What are the
magnitude and direction of the car’s
total displacement?
Vector Addition—Algebraic
Example:
A car moves east 45 km turns and travels
west 30 km. What are the magnitude and
direction of the car’s total displacement?
15 km east
Vector Addition—Algebraic
Example:
A car moves east 45 km turns and travels
north 30 km. What are the magnitude and
direction of the car’s total displacement?
a2 + b2 = c2
c = 54.1 km
magnitude
Vector Addition—Algebraic
Example:
A car moves east 45 km turns and travels
north 30 km. What are the magnitude and
direction of the car’s total displacement?
a2 + b2 = c2
c = 54.1 km
magnitude
Tan  = o/a
Tan  = 30/45
= Tan-1 0.667
 = 33.7
Vector Addition—Algebraic
Example:
A car moves east 45 km turns and travels
north 30 km. What are the magnitude and
direction of the car’s total displacement?
54.1 km
33.7 North of
East (?)
Vector Addition—Algebraic
Example:
A car is driven east 125 km and then
south 65 km. What are the magnitude
and direction of the car’s total
displacement?
Answer:
141 km 27.5 south of east
Vector Addition—Algebraic
Example:
A boat is rowed directly across a
river at a speed of 2.5 m/s. The river
is flowing at a rate of 0.5 m/s. Find
the magnitude and direction of the
boat’s diagonal motion.
Answer
2.55 m/s 11.3 measured from center
of river (or 78.7 measured from
shore)
Vector Resolution
Vector Resolution—the process of
breaking a single vector into its
components
Components—the two perpendicular
vectors that when added together
give a single vector
Components are along the x-axis and
y-axis
Vector Resolution
Example
A bus travels 23 km on a straight
road that is 30 north of east. What
are the east and north components of
its displacement?
Vector Resolution
Example
A bus travels 23 km on a straight road that
is 30 north of east. What are the east
and north components of its displacement?
Vector Resolution
Example
A bus travels 23 km on a straight road that
is 30 north of east. What are the east
and north components of its displacement?
cos  = a/h cos 30 = a/23
a = 23cos30 = 19.9 km east
Vector Resolution
Example
A bus travels 23 km on a straight road that
is 30 north of east. What are the east
and north components of its displacement?
sin  = o/h
sin 30 = o/23
cos  = a/h cos 30 = a/23
o = 23sin30
a = 23cos30 = 19.9 km east
o = 11.5 km north
Vector Resolution
Example
A golf ball, hit from the tee, travels
325 m in a direction 25 south of
east. What are the east and south
components of its displacement?
Answer
East (cos 25 = a/325) 295 m
South (sin 25 = o/325) 137 m
Vector Resolution
Example
An airplane flies at 65 m/s at 31
north of west. What are the north
and west components of the plane’s
velocity?
Answer
north (sin 31 = o/65) 33.5 m/s
west (cos 31 = a/65) 55.7 m/s
Vector Addition and Resolution
Let’s check our knowledge! (6-10)