General Physics (PHYS101)
Vector analysis
Lecture 06
Chapter 3
Outline
•Addition and subtraction of vectors
•Vector decomposition
•Unit vectors
•Dot (scalar) product of vectors
•Cross (vector) product of vectors
2
Adding vectors
•When adding vectors, their directions must be taken
into account.
•Units must be the same.
•Graphical methods
✦Use scale drawings
•Algebraic methods
✦More convenient
3
Adding vectors Graphically
triangle method
y
x
4
Adding vectors Graphically
•Continue drawing the vectors
“tip-to-tail”
•The resultant vector is drawn
from the origin of the first
vector to the and of the second
one.
•Measure the length of the
resultant vector and its angle
5
Adding vectors Graphically
•When you have many vectors,
just keep repeating the process
until all are included
•The resultant is still drawn
from the origin of the first
vector to the end of the last
vector.
6
Alternative Graphical Method
•When you have only two
vectors, you may use the
Parallelogram Method
•All vectors, including the
resultant, are drawn from a
common origin
7
Properties of Vector addition
•Vectors obey the Commutative Law of Addition
•The order in which the vectors are added does not
affect the result
8
Properties of Vector addition
•Vectors also obey the Associativity Law of Addition
•When adding three vectors, it does not matter which
two yo start with
9
Scalar Multiplication of Vectors
•Associative law
•Distributive law
10
Vector Subtraction
y
x
11
Vector Subtraction
•Special case of vector addition
•If A-B, then use A+B:
•Continue with standard vector addition procedure
12
Vector Subtraction
y
𝑫=𝑨−𝑩
x
13
General Physics (PHYS101)
Golibjon Berdiyorov
Building 6, Room 148
Vector analysis
Lecture 06
Chapter 3
Vector Decomposition
•
y
is the projection of the vector
along the x-axis
•
is the projection of the vector
along the x-axis
x
0
•Vector
•Vector
is decomposed to vectors
and
.
is the sum of its components:
•How do we find
and
?
15
y
Unit vectors
• Both
x
0
and
vectors
•The magnitude of the unit vectors
equals 1:
•The vector
y
is expressed as
x
0
16
y
Unit vectors
•The vector
is expressed as
x
0
y
y
x
0
x
0
17
Unit vector in 3D cartesian coordinates
•Unit vector in the directions of
vector
18
Adding and subtracting vectors
Algebraic method
19
Dot product of vectors
•Dot product (or scalar product) of vectors
and
is
defined as
•Dot product is always a scalar quantity
•Two vectors are orthogonal (i.e. perpendicular to each
other) if their dot product is zero
20
Dot products
21
Cross product of vectors
•Cross product is a vector operation that generates a new
vector from the other two vectors.
•The magnitude of cross product of vectors and is
defined as
•Cross product is always a vector perpendicular to the plane.
22
Properties of cross product
•The cross product is anti-commutative since changing the
order of the vectors cross product changes the direction of
the resulting vector
24
Mathematical definition of cross
product
•Two vectors are parallel to each other if and only if:
Cross products
26
Vector analysis
1. The angle between
and the negative y-axis is?
2. A vector in the xy plane has a magnitude of 25 m and
an x component of +12 m and a positive y component.
Find the vector? The angle it makes with the positive y
axis is?
3. If
has the magnitude of 3 m and makes an angle
30o with the +x axis, then the vector
is?
4.
5. A vector
is defined as
Find the magnitude of a vector
if the resultant of
and
is in the y-axis and its magnitude is 5.2.
27
Dot products
1. What is the angle between
and
2. If
what is the angle between them?
28
Cross products
29