This document discusses vectors and their properties. It defines a vector as a physical quantity that has both magnitude and direction, unlike scalars which are only specified by a single value. Velocities and forces are provided as examples of vector quantities. Vectors can be represented graphically as arrows or analytically with symbols and units. Key properties of vectors include how they can be added using the tail-to-head or parallelogram methods of vector addition. Rectangular coordinate systems are also introduced to graph vectors.

1. Vectors

2.  Some physical quantities, such as time,
temperature, mass, and density, can be
described completely by a single number
with a unit.
 But many other important quantities in
physics have a direction associated with them
and cannot be described by a single number.
Vectors

3. Example
A simple example is the motion of an
airplane.
To describe this motion completely, we must
say not only how fast the plane is moving,
but also in what direction.
To fly from Chicago to New York, a plane
has to head east, not south. The speed of the
airplane combined with its direction of
motion together constitute a quantity called
velocity
Vectors

4. Another Example
Another example is force, which in physics
means a push or pull exerted on a body.
Giving a complete description of a force
means describing both how hard the force
pushes or pulls on the body and the
direction of the push or pull.
Vectors

5. “Quantities that can be measured are known
as physical quantities.”
Physical Quantities

7.  A scalar quantity is specified by a single
value with an appropriate unit and has no
direction.
 A vector quantity has both magnitude and
direction.

8. Graphical Representation of aVector
 Graphically a vector is represented by an
arrow op (Fig 2) defining the direction, the
magnitude of the vector being indicated by
the length of the arrow.
 The tail end O of the arrow is called the
origin or the initial point of the vector and
the head P is called the terminal point.

9. Analytically a vector is represented by a
letter with an arrow over it as in Fig.1.
Its magnitude is represented by or A.

10. In printed works, a bold face type A is used to
indicate a vector and for the magnitude of
vector.
A given vector A (or ) can be written as
= A
where A is the magnitude of vector A and so
it has unit and dimension, and
is a dimensionless unit vector with a unity
magnitude having the direction of A.
=

13. Rectangular or Cartesian Coordinate System
Three reference lines drawn perpendicular to
each other are known as coorinate axes and
their point of intersection is known as origin.
This system of coordinate axes is known as
cartesian or rectangular coordinate system as
shown in figure

14. Rectangular or Cartesian Coordinate System
Three dimensional
coordinate system
 Two dimensional
coordinate system
 Always use a right
handed coordinate
system.

15. SOME PROPERTIES OFVECTORS
Equality ofTwoVectors
For many purposes,
two vectors A and B may
be defined to be equal if
they have the same
magnitude and point
in the same direction.
That is, A = B only if A = B and if A and B
point in the same direction along parallel
lines.

16. The Algebra ofVectors
Vector Addition
Addition of two vectors has the simple
geometrical interpretation shown by the
drawing.
The rule is to add B to A, place the tail of B
at the head of A by parallel translation of B.
The sum is a vector from the tail of A to the
head of B.

17. The Algebra ofVectors

18. The Algebra ofVectors
Parallelogram Rule of Addition,
An alternative graphical procedure for adding
two vectors, known as the parallelogram rule
of addition
In this construction, the tails of the two
vectors A and B are joined together and the
resultant vector R is the diagonal of a
parallelogram formed with A and B as two of
its four sides

19. The Algebra ofVectors
Parallelogram Rule of Addition,
Figure 3.9 (a) In this construction, the resultant R is the diagonal of a
parallelogram having sides A and B.
(b)This construction shows that A+B=B+A— in other words, that vector
addition is commutative.

20. The Algebra ofVectors
Associative law of addition:,
Figure 3.10
Geometric constructions for verifying the associative
law of addition.