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Prof. Dasmaya Sidhu delivered this lecture at National Institute of Industrial Engineering for Basic Mechanical Engineering course. It includes: Scalars, Vectors, Operation, Coplanar, Forces, Dot, Product, Parallelogram, Triangle, Addition
Typology: Slides
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triangle law. To express the force and their resultant in coplanar system To introduce the dot product of the vectors and to determine the angle between two vectors
A Vector is represented graphically by an
The length of the arrow represents the
magnitude of the Vector
The “angle
θ
” between the vector and a fixed axis defines the
direction of its line of action
The head of the arrow indicates the
direction of the Vector
Line of action
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θ
Direction
Point “O” is called the tail of the vector while point “P” is called the tip .Vector quantities are represented either by a bold face letters such as
A and its magnitude is
italicized
, or by a character with an arrow on it:
Line of action
θ
” is measured counter clock wise
from the horizontal axis
Equal Vector Two vectors,
and
are equal if they have the same magnitude and direction, regardless of
whether they have the same initial points
Negative Vector
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Negative Vector A vector having the same magnitude as
but in the opposite direction to
is denoted by
, as
shown in Panel
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Null vectors Any vector which has direction but no magnitude is called a null vector. For a null vector itsinitial point and the terminal point will be same. Unit Vector Any vector of unit magnitude is called unit vector.
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To clearly understand how to find the sum of vectors using parallelogram law of vector addition let us taketwo forces of magnitude 10N and 5N represented by the two vectors as shown in the slides below and addthem to find the resultant of the two forces
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Steps to add two vectors using triangle law of vector addition:
Place the two vectors such that the tail of second vector is at the head of the first vector Draw the third side of the triangle. The third side of the triangle is the resultant of the forces with its tail at the tail of thefirst vector and its head at the head of the second vector.
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If the two vectors
and
are collinear, i.e. both have the same line of action, the
parallelogram law reduces to an algebraic or scalar addition:
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A force is a vector quantity since it has a specified magnitude and direction.Forces are
added together or resolved into components using the rules of vector algebra
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and F
2
added together,
according to the parallelogram law, yielding a resultant force F
that form the diagonal of the parallelogram
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R
that form the diagonal of the parallelogram
The law of cosines is useful for finding:
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The third side of a triangle when you know
two sides and the angle between
them
The angles of a triangle when you know
all three sides
What is Angle "C"
2
2
2
c
C
Sin
b
B
Sin
a
A
Sin
=
=
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o
o
y
o
o
y
20
o
o
o
o
o
x
x = 180 - 65 = 115
o
x =
y = 115
o
o
x
o
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o
o
o
o
ϕϕϕϕ
o
R
21
o
o
θ θ
θ θ
o
o
N
F
R
6
.
212
)
115
cos(
150
100
2
150
100
2
2
2
=
−
=
Using Law of cosine
θ θ
θ θ
o
8
.
39
115
sin
6
.
212
sin
150
=
=
θ
θ
Using Law of sine
