Vectors and Vector Addition Methods

Sections of Lecture 1
What is a vector?
• Describing vectors
• Multiplication and Division of vectors
• Addition and Subtraction of Vectors
• Vectors in same or opposite directions
• Vectors at right angles
• All other vectors

Quantity
Vector
(Magnitude &
Direction)
Scalar
(Magnitude)
Quantities like
temperature, time,
mass and
distance are
examples of scalar
quantities.
Quantities like
force, velocity, and
displacement
are examples of
vector quantities.

For example:
35 m is a scalar quantity.
35 m [East] is a vector quantity.

What is a vector?
Vectors are quantities that have magnitude (size) AND
direction.

Describing Vectors
Consider a gravitational force of 65 N [down]
Magnitude unit direction

Important Notation
To describe vectors we will use:
 The bold font: Vector A is A
 Or an arrow above the vector:
 In the pictures, we will always show
vectors as arrows
 Arrows point the direction
 To describe the magnitude of a
vector we will use absolute value
sign: 𝐀 or just A,
 Magnitude is always positive, the
magnitude of a vector is equal to
the length of a vector.

When describing vectors, it is convenient to
use a standard (x,y) reference frame.
+y direction
North
up
-y direction
South
down
+x direction
East
right
-x direction
West
left

In physics, convention dictates vector direction
(angle) is measured from the x axis of the
frame of
reference. +y direction
North
up
-y direction
South
down
+x direction
East
right
-x direction
West
left
30°

In math it is common to describe direction differently:

Some important Vectors:
Equal Vectors:
Vectors having same magnitude &
Direction

Negative Vectors:
Vectors having opposite
Direction

Co-linear Vectors:
Vectors in the same straight line
or parallel

Co-planer Vectors:
Vectors in the same plane

Null Vectors:
A Vector which magnitude is
ZERO
Two equal & opposite forces acting
on a point

If a rope is pulled by two equal forces by two men, then it
remains fixed on that place because of equilibrium forces.
This is null vector

Position Vectors:
Position of any point w. r. t
origin of a reference frame
Y
Z
P
X
O

Displacement vector:
Indicates starting &
ending point, i.e. change
in position

Direction of a Vector
• The direction of a vector is the acute angle it makes with the east-west line. The letter N
or S is written after the measure if the angle followed by the phrase “of E or W.”
• A direction of 60° south of west means that starting from west, you go south by 60°
45°
30°
N
E
S
W
A
B
• The direction of vector A is 30° N of E
• The direction of vector B is 45° S of W

45°
N
E
S
W
50°
20°
B
A
C
D
Specify the directions of
vectors A,B,C, and D Answers:
A is 50° N of E
B is SW
C is 20° S of E
D is S

Operations on Vectors
• Scalar quantities obey the ordinary rules of algebra on addition, subtraction,
multiplication and division. On the other hand,
• Vector quantities are in a different way. The directions must be taken into
consideration when adding, subtracting, and multiplying vectors.

Vector Addition
• Graphical Method
• Parallelogram Method
• Polygon Method
• Analytical Method
• Cosine Law and Sine Law (two vectors are to be added)
• Component Method (more than two vectors)

Vector Addition: Graphical Method
Parallelogram Method (adding two vectors by placing them tail to tail)
• Using a suitable scale, draw the arrows representing the vectors from a
common point.
A
B

Parallelogram Method
• Construct a parallelogram using the two vectors as sides.
A
B

Parallelogram Method
• Draw the diagonal of the parallelogram from the common point. This
represent the resultant, R.
A
B R

Example
Two forces are acting on a particle. Force A is 3.0 N directed north
and the other force B 4.0 N 30 ° N of E. Find the resultant force
using the parallelogram method.

Polygon Method (Tip to Tail Method)
• Represent each vector quantity by an arrow drawn to scale.
• Draw the first arrow observing properly its direction
A

• Connect the tail of the second vector to the head of the first vector.
B
A

• The resultant is the vector that will close the figure by connecting the
vectors. This resultant is drawn from the tail of the first vector to the
head of the last vector.
R
Since the polygon that will be formed is
the triangle, the polygon method of two
vectors is otherwise known as the
triangle method.

Example
Find the resultant of the two forces given in the previous problem
using the polygon method.

Vector Addition: Analytical Method
Part 1: Parallel vectors
For two vectors acting in the same direction- the magnitude of the resultant
is equal to the sum of the magnitudes of the vectors.

Example 1: Blog walks 35 m [E], rests for 20 s and then walks 25 m [E]. What
is Blog’s overall displacement? (what is displacement??)

Solve algebraically by adding the two magnitudes. WE CAN ONLY DO THIS
BECAUSE THE VECTORS ARE IN THE SAME DIRECTION.
SO 35 m [E] + 25 m [E] = 60 m [E]

For two vectors acting in the opposite direction – the magnitude of the
resultant is the difference of the magnitudes of the vectors.

Example 2: Blog walks 35 m [E], rests for 20 s and then walks 25 m [W].
What is Blog’s overall displacement?

Algebraic solution, we can still add the two magnitudes. WE CAN ONLY DO
THIS BECAUSE THE VECTORS ARE PARALLEL
We must make one vector negative to indicate opposite direction.
SO 35 m [E] + 25 m [W]
= 35 m [E] + – 25 m [E]
= 10 m [E]
Note that 25 m [W] is the same as – 25 m [E]

Part 2: Perpendicular Vectors
For two vectors that are perpendicular to each other – the magnitude of the
resultant can be obtained by the Pythagorean theorem.

Example 3: Blog walks 30 m [N], rests for 20 s and then walks 40 m [E]. What
is Blog’s overall displacement?
30 m
40 m

Example 3: Blog walks 30 m [N], rests for 20 s and then walks 40 m [E]. What
is Blog’s overall displacement?

The Pythagorean approach is a useful approach for adding any two vectors
that are directed at right angles to one another. A right triangle has two
sides plus a hypotenuse; so the Pythagorean theorem is perfect for adding
two right angle vectors

• For instance, the addition of three or four vectors does not lead to the
formation of a right triangle with two sides and a hypotenuse.
• If the two vectors that are being added are not at right angles to one
another.

Part 3 - Adding multiple vectors
Consider the following vector addition problem:
A student drives his car 6.0 km, North before making a right hand turn and
driving 6.0 km to the East. Finally, the student makes a left hand turn and
travels another 2.0 km to the north. What is the magnitude of the overall
displacement of the student?

6.0 km N
6.0 km E
2.0 km N
Note: head to tail
Resultant

6.0 km N
6.0 km E
2.0 km N
Resultant
𝑐2 = 𝑎2 + 𝑏2
= 100.0 km 2
𝑐2 = (8.0 km)2 + (6.0 km)2
𝑐2
= 100.0 km 2
𝑐2
c= 10.0 km

Part 3 – Resolutions of Vectors
A single vector is usually divided into two vectors that are perpendicular to
each other. These two vectors are called components and the process of
splitting the vector into its components is called resolution.
The components are normally along the x- and y- axes of the rectangular
coordinate system. The components along the x-axis is called horizontal
component, while the y-axis is called vertical components of vector V

Part 3 – Resolutions of Vectors
as Vx and Vy respectively. In general
𝑽𝒙 = 𝑽𝒄𝒐𝒔𝜃
𝑽𝒚 = 𝑽𝒔𝒊𝒏𝜃
Where 𝜃 is the angle that V makes the x = axis
V
𝑽𝒚
𝑽𝒙
𝜃

Vector Addition: Analytical Methodso
Example:
A person walks 5.0 m in the direction 37° N of E. How far north and how far
east had he walked?
V
V𝒚
V𝒙
37°
V𝒙 = 𝑽𝒄𝒐𝒔𝜃 = 𝟓. 𝟎 𝒎 𝒄𝒐𝒔 37° = 4.0 m
V𝒚 = 𝑽𝒔𝒊𝒏𝜃 = 5.0 m sin 37°= 3.0 m

Quiz
• Two forces are acting on a particle. Force A 3.0 N directed
north and the other force B is 4.0 N 30° N of E. Find the
resultant force using the parallelogram and polygon method.

Vectors and Vector Addition Methods

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