Source: Advexon, Phys.org
M. BALATERO, H. CUBIO, M. EGOT, J.J. LADOR, C. MONTALBAN,
F. SANICO, K. SENADOS, R. SOLIDUM, V. SUAREZ, L.G. TABAR
CHAPTER 0: UNITS, PHYSICAL QUANTITIES AND VECTORS
The primary goal of Physics is to describe an observable natural phenomenon in terms of
the basic concepts and to be able to use these descriptions to predict the outcome of an observation.
Physical quantities are used to describe natural phenomena so it is necessary to identify the
quantities that are used to describe and the relationship between these quantities.
To compare quantities, two things must be specified: a number and the unit of measure,
referred to as magnitude of a physical quantity
In 1971, the 14th General Conference on Weights and Measures defined SI as made up of
seven basic units. See Appendix C.
Aside from these base units there are derived units that are defined from a combination of
base units or with other derived units to describe their relationships in the form of an algebraic
equation. See Appendix.
Unit Consistency and Conversion of Units
Units can be multiplied and/or divided just like ordinary algebraic expressions. This gives
an easy way to convert a quantity from one unit to another to be dimensionally consistent.
Examples:
𝑟𝑒𝑣
a)
To convert 6
to
𝑠
𝑟𝑎𝑑
𝑚𝑖𝑛
Conversion factor to be used:
6
b)
1 rev= 2rad; 1min=60s
rev  2rad   60s 
rad
x
x 
 = 2269.15
min
s
 1rev   1min 
To convert 50
𝑘𝑔
𝑚3
to
𝑔
𝑐𝑚3
Conversion factor to be used: 1 kg = 1000 g; 1m = 100cm
3
3
kg
g  1m 
kg
g
1m
g
50
x 1000
x
= 50
x 1000
x
= 0.05

3
3
3
3
kg  100cm 
kg 1,000,000cm
m
m
cm
c)
x = vt
where
v=5
m
and
𝑡 = 20𝑚𝑖𝑛
s
If x is measured in meters, then the product vt must also be expressed in meters.
To solve, first convert the unit of t from minutes to seconds using the conversion
factor 1min=60s. Then multiply to the value of v.
20 min x 60
s
= 1200 s
min
x= 5
m
x 1200 s = 6000 m
s
The Scientific Notation
Physical quantities vary from very large numbers to very small numbers. A more convenient
and compact way of writing these values uses the powers of ten notation, exponential notation or
scientific notation wherein one can determine the number of significant digits immediately as
1
well as the place value of the digit. Prefixes are used to denote these place values. See Appendix
B for the standard SI prefixes.
Format:
C.MMMMM x 10e
where:
C
M
10
e
– the characteristic digit, may be any digit from 1 to 9
– the mantissa digits, may be any digit from 0 to 9
– the base,
– the exponent, the number of times the decimal is moved to either towards left
or right.
Rule 1: Positive exponent results when the decimal point is moved from right to left direction.
Example: 98067.5321 = 9.80675321 x 10 4 = 9.81 x 104
Rule 2: Negative exponent results when the decimal point is moved from left to right direction.
Example: 0.000980675321 = 9.80675321 x 10 -4 = 9.81 x 10
-4
Significant Figures
The number of significant figures is the number of digits about which we have some degree
of certainty. It is a measure of the degree of reliability of a certain measurement.
Some of the rules in determining the Number of Significant Figures:
1. All nonzero digits are significant.
Example: 3.1416
→
5 significant figure
2. All zeros between nonzero digits are significant.
Example: 5.0046
→
5 significant figures
3. All zeros before the first nonzero digit are not significant
Example: 0.0001
→
1 significant figure
4. All zeros to the right of the last nonzero digit are significant.
Example: 77.800
→
5 significant figure
In adding or subtracting quantities, the number of decimal places in the answer should be
the same as the least number of decimal places in any of the numbers being added or subtracted.
Example:
5.67 m
+ 1.1 m
+ 0.9378 m
7.7 m
(two decimal places)
(one decimal place) – least number of decimal places
(four decimal places)
(one decimal place)
In multiplication and division, the number of significant figures in the answer is the same
as the number of significant figures in the input number that has the fewest.
Example:
2
11.63cm
x 5.74cm
66.8cm2
(4 significant figures)
(3 significant figures) – least number of S.F.
(3 significant figures)
Scalar and Vector Quantities
Scalar Quantity – physical quantity that is completely described by its magnitude only; hence, it
can be operated ordinarily using the four fundamental arithmetic operations.
Vector Quantity – physical quantity that is completely described by a magnitude and direction
– denoted usually by an alphabet with arrow over it to indicate its direction.
Example:

A
– graphically represented by scaled line with an arrow at the
tip and a label:
Methods of Adding Vector Quantities
Resultant,
:

R - total or vector sum of two or more vectors.
   
R = A + B + C + ...
1.
Graphical Method:

-measured using ruler
R
R -measured using protractor
Magnitude of
Direction of
Types:
a.
Parallelogram Method – a line is drawn parallel to the given
vector whose length is equal to the vector. The resultant
vector is drawn from the origin to the tip of the intersection
of the parallel lines and the resultant direction is measured
from the horizontal.
b.
Polygon Method – head-to-tail connection of vectors and the
resultant is connected from the tail of the first vector to the head of
the last vector added. It is more convenient to use when adding more
than two vectors.
2.
Analytical Method
a.
Law of Cosine and Sine Method - requires basic Trigonometry
knowledge in adding two vectors
•
Magnitude of

R - obtained using Law of Cosine
R =
A2 + B 2 − 2 AB cos 
 
 is the angle between A & B

Direction of R - obtained using Law of Sine
where,
•
3
 B sin  

 R 
sin  R sin 
=
B
R
 R = sin −1 
For special case when two vectors are perpendicular to each other:
Magnitude of

R - obtained using Pythagorean Theorem
R 2 = A2 + B 2 ---> R = A2 + B 2
Direction of

R - obtained using
B
 R = tan −1  
 A
b. Component Method – each vector is resolved into its components
y-axis
Components
Vectors
𝐴⃑
X
⃑⃑
𝐵
𝐶⃑
𝑅⃑⃑
Y
⃑⃑
𝐵
Ax = A cos
Ay = Asin 
Bx = − B cos 
C x = C cos 
Rx = Ax + Bx + C x
B y = B sin 

Magnitude of R : R =
Rx + R y
2
2
𝐴⃑
x-axis
C y = −C sin 
R y = Ay + B y + C y
Direction of

R
:
𝐶⃑
 Ry
 Rx
 R = tan −1 



Unit Vector
has a magnitude of 1 and points in a particular
direction of a vector in space.
iˆ − points in the positive x-axis direction
ĵ
k̂
− points in the positive y-axis direction
− points in the positive z-axis direction
Vector sum in terms of unit vectors

A = Axiˆ + Ay ˆj + Az kˆ

B = Bxiˆ + By ˆj + Bz kˆ
  
R = A + B = ( Ax + Bx )iˆ + ( Ay + By ) ˆj + ( Az + Bz )kˆ
Product of Vectors
1. Scalar or Dot Product – the resulting product is a scalar quantity.
Equations:
a)
 
A  B = AB cos
(Definition)
(1-1)
where,  = angle between the two vectors
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b)
 
A  B = Ax Bx + Ay B y + Az Bz
Example:
 
Application: Work, W = F  s
(Component)
iˆ  iˆ = 1
(1-2)
ˆj iˆ = 0
2. Vector or Cross Product – the resulting product is a vector quantity.
The magnitude of resulting product is obtained using
a)
⃑⃑ = 𝐴𝐵𝑠𝑖𝑛∅
𝐴⃑ × 𝐵
(Definition)
b)
⃑⃑ = 𝐶𝑥 + 𝐶𝑦 + 𝐶𝑧
𝐴⃑ × 𝐵
(1-3)
(Component)
(1-4)
where,
C x = ( Ay B z − Az B y )iˆ
C y = ( Az B x − Ax B z ) ˆj
C = ( A B − A B )kˆ
z
c)
⃑⃑ =
𝐴⃑ × 𝐵
iˆ
Ax
Bx
x
ˆj
Ay
By
y
y
x
kˆ
Az
Bz
(Determinant form)
= ( Ay B z − Az B y )iˆ + ( Az Bx − Ax Bz ) ˆj + ( Ax B y − Ay Bx )kˆ
(1-
5)
The direction of resulting product is obtained using the
⃑⃑ , place your right hand along the length of
Right- Hand Rule - In 𝐴⃑ × 𝐵

toward B .

A
and curl your
fingers
Your thumb then points to the direction of the
resulting product vector.
Example: 𝑖̂ × 𝑖̂ = 0
𝑖̂ × 𝑗̂ = 𝑘̂
Application: Torque, 𝜏⃗ = 𝑟⃑ × 𝐹⃑
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SAMPLE PROBLEMS:
University Physics 11th Ed., p. 28
1. Determine the resultant of the following two vectors:
𝐴⃗ = 6 𝑚, 𝐸
⃑⃗ = 3 𝑚 𝑎𝑡 50° 𝑆 𝑜𝑓𝐸
𝐵
using: a) Graphical Method
b) The Law of Sine and Cosine Method
c) Component method
Solution:
Scale: 1 unit = 1 m
a) Graphical Method (polygon)
Using ruler for the magnitude
and protractor for the direction:
𝑅⃑⃑ = 8𝑚, 160 S of E
b) Law of Cosine and Sine Method
Cosine Law (Magnitude):
R 2 = A 2 + B 2 − 2 AB cos 
R = 62 + 32 − 2(6)(3) cos 130 =
8.25m
Sine Law (Direction):
sin  R sin 1300
=
B
R
B

 R = sin −1  sin 130 0  = 15.65
R

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c) Component Method
x - Components:
Ax = 6.0m cos 0 0 = 6.0m
Bx = 3.0m cos 500 = 1.93m
y - Components:
Ay = 6.0m sin 0 0 = 0
B y = 3.0m( − sin 50 0 ) = −2.30m
R x = Ax + B x = 7.93m
R y = Ay + B y = −2.30m
R=
(magnitude)
Rx + R y =
2
2
Ry
 R = tan −1 (

Hence, R
=
Rx
8.26m
) = -16.17° (direction)
8.26m, 16.17° South of East
2. Given two vectors:
𝐹⃗1 = 2𝑖̂ − 𝑗̂ and 𝐹⃗2 = −𝑖̂ + 3𝑗̂
Determine:
 
+ F2


b) F1 − 2F2
 
c) F1  F2
 
d) F1  F2
a) F1
Solution:
a)
b)
c)
 
F1 + F2 = 2iˆ − ˆj + ( −iˆ) + 3 ˆj = iˆ + 2 ˆj


F1 − 2 F2 = 2iˆ − ˆj − 2(−iˆ + 3 j ) = 4iˆ − 7 ˆj
 
F1  F2 = (2)(-1) + (-1)(3) = - 2 - 3 = -5
Note: try finding the dot product using equation (1-1)

d) F1





 F2 = [(-1)(0) - (0)(3)] i + [(0)(-1) - (2)(0)] j + [(2)(3) - (-1)(-1]k = 5k
Note: try finding the cross product using equations (1-3) and (1-5)
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