V ECTOR A NALYSIS
Alipar, Emmanuel B.
BSECE - 3
I. What is Vector?
II. Application of Vector Analysis
●Divergence of Vector Field
it has a quantity that has direction as well as
Tip / Arrowhead
● Commutative: A + B = B + A
● Associative: A + (B + C) = (A + B)=C
● Distributive: K (A + B)=KA + KB
● Anti Commutative: A x B = B x A but A x B = -B x A
● It is not an Associative: A x (B x C) = (A x B) x C
● It is a Distributive: A x (B + C) = A x B + A x C
1. Electromagnetism (Gauss's
is a law relating the distribution of electric charge to
the resulting electric field. Gauss's law states that:
electric flux through any closed surface is
proportional to the enclosed electric charge.
provides a relationship between the net electrical
flux Φ through a closed surface and the net charge
enclosed by that surface. Gauss’ law states that
the net electric flux out of a closed surface is equal
to the charge enclosed by the surface divided by
Combination of Gauss's Law &
Divergence of Vector field
Electromagnetic Application (Gauss's
In Electromagnetism, one of its application is
the Gauss's Law which is combined with
divergence theorem and surface integrals of a
Divergence of the Vector Fields
is a vector operator that measures the
magnitude of a vector field's source or sink
at a given point, in terms of a signed
scalar. More technically, the divergence
represents the volume density of the
outward flux of a vector field from an
infinitesimal volume around a given point.
Divergence of the vector field
* The direction of the arrow indicates the flow source of
outward and inward flux.
2. Signal Processing
it is an area of systems engineering, electrical
engineering and applied mathematics that deals with
operations on or analysis of analog as well as digitized
signals, representing time-varying or spatially varying
Signals of interest can include sound, electromagnetic
radiation, images, and sensor readings, for example
biological measurements such as electrocardiograms,
control system signals, telecommunication
transmission signals, and many others.
Example of Signal Processing
Consider the problem of designing a smooth
takeoff path for an airplane. The initial vector is
the plane heading down the runway just before
takeoff. The final vector is the level flight path at
a given altitude. You must design the path the
plane takes between these two vectors while
avoiding jarring the passengers too much
during the takeoff.
Example of Signal Processing (Cont.)
1. Idealization: We will idealize the aircraft as a particle. We
can do this because the aircraft is not rotating during takeoff.
2. FBD: The figure shows a free body diagram. represents
the (unknown) force exerted on the aircraft due to its
3. Kinematics: We must calculate the acceleration required
to reach takeoff speed. We are given (i) the distance to
takeoff d, (ii) the takeoff speed and (iii) the aircraft is at rest
at the start of the takeoff roll. We can therefore write down
the position vector r and velocity v of the aircraft at takeoff,
and use the straight line motion formulas for r and v to
calculate the time t to reach takeoff speed and the
acceleration a. Taking the origin at the initial position of the
aircraft, we have that, at the instant of takeoff
Example of Signal Processing (Cont.)
The Vector Transition plot shows one possible
smoothed trajectory (in blue) compared to the direct
path (in red). The blue trace seems sufficiently gradual
to make a good choice. How was this path computed?
Look at the Second Derivative (Acceleration) plot,
which has two back-to-back Hann windows (hann) .
The position vector of point P is the directed
distance from the origin O to P.
rP = OP =xax + yay + zaz
rPQ = rQ – rP
= (xQ – xP)ax + (yQ – yP)ay + (zQ - zP)az
Illustration of Position Vector
rP = xax + yay + zaz