Physics for Engineers 12/3/2021

Vectors and Scalars

Physics for Engineers

Learning Objectives

At the end of the lesson, students will be able to:

1. Differentiate between vectors and scalar quantities.
2. Determine quantities that are appropriate to be represented by
a vector and represent these vectors geometrically.
3. Name vectors in component form and use them to solve
problems involving addition and subtraction.
4. Solve problems involving quantities that can be represented
by vectors.
5. Solve vector problem using component method

PHY 032 1
Physics for Engineers 12/3/2021

Physics is a science that deals with matter

and energy and their interaction.
 Foundation on which all modern science and
technology.
 Today’s medical imaging technology, computers,
telecommunication is yesterday’s physics
research.
 Teaches a particular way of addressing
problems and observing the environment.
 Helps you understand what is possible and what
is not, which information to believe and which
not.
3

Branches of Physics

 Mechanics
 Sound and Wave Motion
 Thermodynamics
 Electricity and Magnetism
 Optics
 Modern Physics

PHY 032 2
Physics for Engineers 12/3/2021

UNITS
Systeme Internationalle (SI), which is a version of the
metric system. The “standard units:”

Conversion of Units

PHY 032 3
Physics for Engineers 12/3/2021

Conversion of Units

Unit Conversion

PHY 032 4
Physics for Engineers 12/3/2021

Rules of Conversion of Units

1. Subtract Exponents
2. Move decimal that amount in the direction of
the wanted unit.

Example

1. 100 L → μL 2. 4000 mm → km
10 10 10 10
0 − −6 = 6 −3 − 3 = −6
From the decimal point of From the decimal point of 4000,
100, move to the right move to the left about 6 steps
about 6 steps 0.004000.
100.000000 4000𝑚𝑚 = 0.004 𝑘𝑚
100𝐿 = 100,000,000𝜇𝐿

10

PHY 032 5
Physics for Engineers 12/3/2021

Example

Convert: 350 kg/m → g/cm

1𝑘𝑔 = 1000 𝑔
1 𝑚 = 100 𝑐𝑚

 350 × ×

 350 × ×
,
 = 0.35
, ,
11

Example
Convert: 240 mi/hr → km/s
1 𝑚𝑖 = 1.609 𝑘𝑚
1 ℎ𝑟 = 3600 𝑠
.
 240 × ×
.
 = 0.107 𝑘𝑚/𝑠

12

PHY 032 6
Physics for Engineers 12/3/2021

Scalar
Scalar Magnitude
A SCALAR is ANY Example
quantity in physics that Speed 20 m/s
has MAGNITUDE, but
NOT a direction
associated with it. Distance 10 m
Magnitude – A numerical
value with units. Age 15 years

Heat 1000
calories

13

Vector

A VECTOR is ANY Vector Magnitude

quantity in physics that & Direction
has BOTH Velocity 20 m/s, N
MAGNITUDE and
DIRECTION. Acceleration 10 m/s/s, E
Force 5 N, West

Vectors are typically illustrated by

drawing an ARROW above the symbol.
The arrow is used to convey direction
and magnitude.

14

PHY 032 7
Physics for Engineers 12/3/2021

1. A car traveling 40 mph.

2. A motorcycle traveling 60 mph due north.

3. A train traveling 40mph east to the beach.

4. A child’s weight on a scale.

15

1. A boat traveling 50 mph 20°east of north

2. An object falling straight down at 15 mph
3. A worker pushing an object with a force of 30 newtons

#3 There is a magnitude, but no direction!

16

PHY 032 8
Physics for Engineers 12/3/2021

Vectors are used to represent vector quantities on a diagram. A vector is

composed of a line segment drawn to scale with an arrowhead at one end.
The tail of the vector is at its origin and the tip is at the terminal point
(arrowhead). The length of the vector represents its magnitude and the
arrowhead indicates its direction.

B
Initial Point Terminal Point
or tail 𝑎⃗ or tip

17

Collinear Vectors

 are vectors that exist in the same dimension.

 they exist either in the same direction or in
the opposite direction.

Equivalent vectors
Are vectors with the same magnitude
and direction 18

PHY 032 9
Physics for Engineers 12/3/2021

Non-collinear Vectors

 are vectors that exist in more than one

dimension.
 They are located along different straight lines.

19

Vector direction

v v

α
v

True Bearing
is always
measured
Quadrant bearing clockwise from the Standard Position
v is between 0° and 90° east north-south line. is measured counter
or west of the north-south line. True bearings are clockwise from 0°.
v is S70°E always given in
three digits. 050°
20

PHY 032 10
Physics for Engineers 12/3/2021

Angles in Standard Position

 An angle on the
coordinate plane is in
standard position if the
vertex is at the origin and
one ray is on the positive
x-axis
 The ray on the x-axis is
called the initial side of
the angle
 The ray that rotates about
the center is called the
terminal side.

Vector Direction: Standard Position

 If the measure of an
angle is positive, the
terminal side is rotated
counter clockwise.

 If the measure of an
angle is negative, the
terminal side is rotated
clockwise.

PHY 032 11
Physics for Engineers 12/3/2021

Vector Direction: Quadrant Bearing

 The quadrant bearing

system divides the
compass into four
equal sections of 90
degrees.
 The bearing of a line is
measured as an angle
from the reference
meridian, either the
north or the south, and
toward the east or the
west.

Vector Direction: Quadrant Bearing

 N22°W = 22° West of North

 N40°E = 40° East of North
 S36°W = 36° West of South
 S71°E = 71° East of South

PHY 032 12
Physics for Engineers 12/3/2021

Vector Direction: Quadrant Bearing

N N
NW NE
W of N E of N
N of E
N of W
W E W E
S of W S of E

W of S E of S SE
SW
S S

25

Vector Direction: True Bearings

True bearings are equivalent to the angle measured
clockwise from north. It is always measured in 3 digits.

Ex. The bearing of Karen from Stephen

can be described as 063°.

Ex. The bearing of Karen from Stephen

can be described as 216°.

26

PHY 032 13
Physics for Engineers 12/3/2021

Vector Direction: Name the direction of vector r, 3 different ways.

r Standard Position: 120°

Quadrant Bearing: N30°W

True Bearing: 330°

27

Unit Vector Notation

An effective and popular system used in engineering is
called unit vector notation. It is used to denote
vectors with an x-y Cartesian coordinate system.

28

PHY 032 14
Physics for Engineers 12/3/2021

Unit Vector Notation

=3j
J = vector of magnitude “1” in the “y” direction

= 4i
i = vector of magnitude “1” in the “x” direction

The hypotenuse in Physics is

called the RESULTANT or
VECTOR SUM.

The LEGS of the triangle are

A  4iˆ  3 ˆj called the COMPONENTS
3j
Vertical Component

NOTE: When drawing a right triangle that

4i conveys some type of motion, you MUST draw
Horizontal Component your components HEAD TO TOE.

29

Unit Vector Notation

iˆ - unit vector  1 in the  x direction The proper terminology is to use
the “hat” instead of the arrow. So
ˆj - unit vector  1 in the  y direction we have i-hat, j-hat, and k-hat
which are used to describe any
kˆ - unit vector  1 in the  z direction type of motion in 3D space.

How would you write vectors J and K in

unit vector notation?

J  2iˆ  4 ˆj

K  2iˆ  5 ˆj

30

PHY 032 15
Physics for Engineers 12/3/2021

Applications of Vectors
VECTOR ADDITION – If 2 similar vectors point in the SAME
direction, add them.

 Example: A man walks 54.5 meters east, then another 30

meters east. Calculate his displacement relative to where he
started?
54.5 m, E + 30 m, E Notice that the SIZE of
the arrow conveys
MAGNITUDE and the
84.5 m, E way it was drawn
conveys DIRECTION.

31

Applications of Vectors

VECTOR SUBTRACTION - If 2 vectors are going in

opposite directions, you SUBTRACT.

 Example: A man walks 54.5 meters east, then 30

meters west. Calculate his displacement relative to
where he started?
54.5 m, E
-
30 m, W

24.5 m, E

32

PHY 032 16
Physics for Engineers 12/3/2021

Non-Collinear Vectors
When 2 vectors are perpendicular, you must use
the Pythagorean theorem.
c2  a2  b2
The hypotenuse in Physics Finish
is called the RESULTANT.
c  a2  b2
𝐵
Vertical
Component
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
qHorizontal Component 𝜃 = tan
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝐴⃗
Start
The LEGS of the triangle are called the COMPONENTS

33

A man walks 95 km, East then 55 km, north. Calculate his

RESULTANT DISPLACEMENT.

. c2  a 2  b2  c  a 2  b2
109.8 km
55 km, N c  Resultant  952  552
c  12050  109.8 km
q
95 km,E opposite side
q  tan 1
adjacent side
q  tan 1 (55 / 95)  30 o

So the COMPLETE final answer is : 109.8 km, 30° North of East

109.8 km, 30°
109.8 km, 060°
𝟗𝟓 ̂ 𝒌𝒎 + 𝟓𝟓 ̂ 𝒌𝒎

34

PHY 032 17
Physics for Engineers 12/3/2021

Example
A bear, searching for food wanders 35 meters east then 20 meters north.
Frustrated, he wanders another 12 meters west then 6 meters south. Calculate
the bear's displacement.
23 m, E
- =

12 m, W
- =
14 m, N
6 m, S
20 m, N
R  14 2  23 2  26.93m
14
35 m, E R 14 m, N q  tan 1
q
23
q  31.3o
23 m, E
The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST

35

Example
A boat moves with a velocity of 15 m/s, N in a river which
flows with a velocity of 8.0 m/s, west. Calculate the
boat's resultant velocity with respect to due north.

Rv  8 2  15 2  17 m / s
8.0 m/s, W
8
15 m/s, N
q  tan 1
Rv q 15
q  28.1o

The Final Answer : 17 m/s, @ 28.1 degrees West of North

36

PHY 032 18
Physics for Engineers 12/3/2021

What if you are missing a component?

The goal: ALWAYS MAKE A RIGHT
TRIANGLE!

V.C = ? To solve for components, we often use

𝑨
the trig functions since and cosine.
q
H.C. = ?

Horizontal Component:
𝐴 = 𝐴 cos 𝜃
Vertical Component:
𝐴 = 𝐴 sin 𝜃

37

Example
A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate
the plane's horizontal and vertical velocity components.

Ax  63.5 cos 32  53.85 m / s

H.C. =? Ay  63.5 sin 32  33.64 m / s
32
V.C. = ?

63.5 m/s
𝑚
𝐴 = 53.85 ,𝐸
𝑠
𝑚
𝐴 = 33.64 , 𝑆
𝑠

38

PHY 032 19
Physics for Engineers 12/3/2021

Example
A storm system moves 5000 km due east, then shifts course at 40
degrees North of East for 1500 km. Calculate the storm's
resultant displacement.
Ax  1500 cos 40  1149.1 km, E
1500 km
V.C. Ay  1500 sin 40  964.2 km, N
40
5000 km, E H.C.

R  6149.12  964.2 2  6224.14 km

5000 km + 1149.1 km = 6149.1 km 964.2
q  tan 1
6149.1
q  8.91o
R
964.2 km
q
6149.1 km The Final Answer: 6224.14 km @ 8.91
degrees, North of East

39

Components of Vector


H.C. Components of a Vector. The original vector, defined

relative to a set of axes. The H.C. stretches from the
start of the vector to its furthest x-coordinate. The V.C.
V.C. stretches from the x-axis to the most vertical point on
the vector. Together, the two components and the
vector form a right triangle.

40

PHY 032 20
Physics for Engineers 12/3/2021

Adding vectors using Analytical

Method (COMPONENT Method)


41

90°
0,1
y

Quadrant II: Quadrant I:

−𝑥, 𝑦 𝑥, 𝑦
−cos 𝜃 , sin 𝜃 cos 𝜃 , sin 𝜃

180° −1,0 x 1,0 0°

Quadrant III: Quadrant IV:

−𝑥, −𝑦 𝑥, −𝑦
− cos 𝜃 , − sin 𝜃 cos 𝜃 , −sin 𝜃

0, −1
270°
42

PHY 032 21
Physics for Engineers 12/3/2021

Example: Find the components of

vectors. y
𝐴 = 𝐴 cos 𝜃 = −5 cos 30 = −4.33 N
𝐵 =2𝑚
𝐴 = 𝐴 sin 𝜃 = −5 sin 30 = −2.5 N
15°

− cos 𝜃 , sin 𝜃
x
30°
𝑁𝑜𝑡𝑒: The angle 𝜃 is between the line and the 𝑥-axis.
So, the angle 𝜃 must be equal to 75° for vector B.
𝐵 = 𝐵 cos 𝜃 = −2 cos 75 = −0.518 m
𝐵 = 𝐵 sin 𝜃 = 2 sin 75 = 1.932 𝑚
𝐴=5𝑁
− cos 𝜃 , − sin 𝜃

𝐹 = 𝐹 cos 𝜃
𝐹 = 𝐹 sin 𝜃 43

Example


1500 km

40
5000 km, E

Vecto X-component Y-component

r
A 5000 cos 0 = 5000 5000 sin 0 = 0
B 1500 cos 40 = 1149.1 1500 sin 40 = 964.2

44

PHY 032 22
Physics for Engineers 12/3/2021

Example


Vecto X-component Y-component

r
A 5000 cos 0 = 5000 5000 sin 0 = 0
B 1500 cos 40 = 1149.1 1500 sin 40 = 964.2

The Final Answer: 6224.14 km @ 8.91

degrees, North of East
45

Example


Vectors x-component y-component

A 35 cos 0 = 35 35 sin 0 = 0
B 20 cos 90 = 0 20 sin 90 = 20
C 12 cos 180 = -12 12 sin 180 = 0
D 6 cos 270 = 0 6 sin 270 = -6

The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST

46

PHY 032 23
Physics for Engineers 12/3/2021

Example

Four vectors, A, B, C and D, are shown in the

figure. The sum of these four vectors is a vector
having magnitude and direction.

47

Solution
Vectors x-component y-component
A 6 cos 90 = 0 6 sin 90 = 6
B -4 cos 30 = -3.5 -4 sin 30 = -2
C 6 cos 270 = 0 6 sin 270 = -6
D 4 cos 30 = 3.5 -4 sin 30 = -2

The Final Answer: 4 cm, South

48

PHY 032 24
Physics for Engineers 12/3/2021

Example


F2

R=8N
F1= 5 N

49

Why Rx = 0? Because the resultant is a vertical vector.

No Horizontal vector

Rx = F1x + F2x Ry = F1y + F2y

0 = 5 cos 45 + F2x 8 = 5 sin 45 + F2y
F2y = 8 - 5 sin 45 = 4.46 N
F2x = -5 cos 45 = -3.53 N

F2

R=8N
F1= 5 N

50

PHY 032 25
Physics for Engineers 12/3/2021

Example
Two forces with magnitudes of 15 pounds and 35 pounds
are applied to an object. The magnitude of the resultant is
28 pounds. Find the measurement of the angles between
the resultant vector and the vector of the 15 pound force to
the nearest whole degree.

51

52

PHY 032 26
Physics for Engineers 12/3/2021

Adding vectors using Graphical

Method

53

Graphical Method
Steps:
1. Draw the three displacement vectors
2. Place the vectors head to tail retaining
both their initial magnitude and
direction.
3. Draw the resultant vector, R.
4. Use a ruler to measure
the magnitude of R, and
a protractor to measure the
direction of R.

54

PHY 032 27