General Physics 1
(Phys 110 : Mechanics)
CHAPTER 3
Vectors
Phys
110
Lesson 1 of 5
Slide 1
Chapter 3 : Vectors
Objectives covered in this lesson :
1.
Vectors:
Definitions of
to define vector quantity and scalar quantity
and differentiate between them.
2.
vectors and scalars.
to add vectors geometrically and write the
resultant equation.
Adding Vectors
geometrically.
3.
to identify vector addition properties:
commutative law, associative law and vector
subtraction.
4.
to find the inverse of any vector.
Phys
110
Chapter 3 : Vectors
Lesson 1 of 5
Slide 2
3-1 What is Physics ?
We use “the language of vectors” in our daily lives to describe directions.
3-2 Vectors and Scalars :
In one dimensional motion:
The (+) or (-) sign of vector quantities indicate their direction.
In 2D and 3D motions:
We have to use a vector ( ‫ سهم في الرسم‬, 𝒂 ) to describe the direction.
Phys
110
Lesson 1 of 5
Slide 3
Chapter 3 : Vectors
3-2 Vectors and Scalars :
Vector Quantity
Scalar Quantity
is a quantity which has both
magnitude and direction.
is a quantity which has only a
magnitude and no direction.
it can be represented by a vector.
-
Vectors are combined with certain
rules.
scalars are combined with the rules
of ordinary algebra.
Examples:
Examples:
displacement, velocity, acceleration.
time, mass, energy, temperature.
Q: is (a temperature of -5ºC) a vector or a scalar?
Phys
110
Chapter 3 : Vectors
3-2 Vectors and Scalars :
The simplest vector quantity:
“Displacement”
A vector representing “displacement” is called a:
displacement vector
The displacement from A to B is represented by:
an arrow from A to B
The length of the arrow: is proportional to the displacement
magnitude.
The direction of the arrow: indicated the displacement direction.
Lesson 1 of 5
Slide 4
Phys
110
Lesson 1 of 5
Slide 5
Chapter 3 : Vectors
3-2 Vectors and Scalars :
The three arrows: (from A to B, from A' to B', and from A'' to B'‘)
have the same magnitude and direction.
A vector can be shifted without changing its value if its length and
direction are not changed.
In books, vectors are written in two ways:
Method 1:
Method 2:
𝑎
a
(using an arrow above)
(using boldface print)
In our book, graphically,
the vector arrow is drawn
with a triangle head, to
distinguish it from other
arrows.
The magnitude of the vector is indicated by italic print: a.
Phys
110
Chapter 3 : Vectors
3-3 Adding Vectors Geometrically :
A particle moves from A to B and then later from B to C:
It undergoes two displacements, represented by two vectors:
1. A displacement vector 𝑎 from A to B.
2. A displacement vector 𝑏 from B to C.
Its overall displacement, is represented by one vector:
A net displacement vector 𝑠 from A to C.
𝒔= 𝒂+ 𝒃
Vector Addition
Q: does that mean:
magnitude of 𝑠 = magnitude of 𝑎 + magnitude of 𝑏 ?
Lesson 1 of 5
Slide 6
Phys
110
Chapter 3 : Vectors
3-3 Adding Vectors Geometrically :
How to draw 𝒂 , 𝒃 , 𝒂𝒏𝒅 𝒔:
1. Sketch vector 𝑎 using an appropriate scale, in the
appropriate angle.
2. Sketch vector 𝑏 using the same scale, in the appropriate
angle.
3. Place the tail of 𝑏 at the tip (head) of 𝑎.
4. The vector sum 𝑠 starts from the tail of 𝑎 to the tip of 𝑏.
Lesson 1 of 5
Slide 7
Phys
110
Chapter 3 : Vectors
3-3 Adding Vectors Geometrically :
Comutative Law:
It means that: the order of addition does not matter.
Lesson 1 of 5
Slide 8
Phys
110
Chapter 3 : Vectors
Lesson 1 of 5
Slide 9
3-3 Adding Vectors Geometrically :
Associative Law:
It means that:
when there are more than two vectors, we can group them in any order as we add them.
Phys
110
Chapter 3 : Vectors
Lesson 1 of 5
Slide 10
3-3 Adding Vectors Geometrically :
The vector −𝑏 is a vector with the same magnitude as 𝑏 but is in the opposite direction.
Vector subtraction:
It means that:
We find the difference vector 𝑑 by adding the vector −𝑏 to the vector 𝑎.
Phys
110
Chapter 3 : Vectors
3-3 Adding Vectors Geometrically :
How to draw “Vector Subtraction”:
1. Sketch vector 𝑎 using an appropriate scale, in the
appropriate angle.
2. Sketch vector 𝑏 using the same scale, in the
appropriate angle.
3. Vector −𝑏 is: swap the head of vector 𝑏 with its tail.
4. Place the tail of −𝑏 at the tip of 𝑎.
5. The difference vector 𝑑 starts from the tail of 𝑎 to the
tip of −𝑏.
Lesson 1 of 5
Slide 11
Phys
110
Chapter 3 : Vectors
Lesson 1 of 5
Slide 12
3-3 Adding Vectors Geometrically :
We can move a term from one side of a vector equation to the other, but we must
change its sign (like normal algebra):
Q: can we do this:
(displacement vector) + (displacement vector) ?
(displacement vector) + (velocity vector) ?
Phys
110
Chapter 3 : Vectors
Lesson 1 of 5
Slide 13
3-3 Adding Vectors Geometrically :
A:
(a) 7 m ( 𝑎 and 𝑏 are in the same direction).
(b) 1 m ( 𝑎 and 𝑏 are in opposite directions).
Phys
110
Chapter 3 : Vectors
Sample Problem (3-1) :
Lesson 1 of 5
Slide 14
Phys
110
Lesson 1 of 5
Slide 15
Chapter 3 : Vectors
Sample Problem (3-1) :
B
Draw each vector alone
A
Decide on the scale
Phys
110
Chapter 3 : Vectors
Sample Problem (3-1) :
Lesson 1 of 5
Slide 16
Phys
110
Lesson 1 of 5
Slide 17
Chapter 3 : Vectors
Sample Problem (3-1) :
C
Sum vectors on the same axis
Phys
110
Lesson 1 of 5
Slide 18
Chapter 3 : Vectors
Sample Problem (3-1) :
D
Sum remaining vectors with them
Drawings that give longest vector
𝒂 + −𝒄 + (𝒃)
−𝑏
Finish
𝑏
OR
𝒂 + −𝒄 + (−𝒃)
Start
(−𝒃) + 𝒂 + −𝒄
−𝑏
𝑏
OR
Finish
𝒃 + 𝒂 + −𝒄
Start
Note:
𝒃 + 𝒂 + −𝒄 = 𝒂 + −𝒄 + (𝒃)
Phys
110
Chapter 3 : Vectors
Lesson 1 of 5
Slide 19
Sample Problem (3-1) :
So, the resultant vector from all three displacements that gives the
greatest distance is 𝑑 where:
𝒅 = 𝒃 + 𝒂 + −𝒄 = 𝒂 + −𝒄 + (𝒃)
The magnitude of 𝑑 is measured (from the graph by the ruler) to be:
𝒅 = 𝟒. 𝟖 𝒌𝒎
The direction of 𝑑 is measured (from the graph by a ‫ )منقلة‬to be at an angle ( ) north of
east.
Phys
110
Lesson 1 of 5
Slide 20 (last)
Chapter 3 : Vectors
Summary:
Vectors:
Next lesson will cover:
Definitions of vectors and scalars.
Section (3-4).
Addition of vectors graphically.
Sample problem (3-2).
Section (3-5).
Any Questions?