Mathematicians and scientists call a quantity which depends on direction a vector quantity, and a quantity which does not depend on direction is called a scalar quantity. Vector quantities have two characteristics, a magnitude and a direction. Scalar quantities have only a magnitude. When comparing two vector quantities of the same type, you have to compare both the magnitude and the direction. For scalars, you only have to compare the magnitude. When doing any mathematical operation on a vector quantity (like adding, subtracting, multiplying...) you have to consider both the magnitude and the direction. This makes dealing with vector quantities a little more complicated than scalars. On the slide we list some of the physical quantities discussed in the Beginner's Guide to Propulsion and group them into either vector or scalar quantities. Of particular interest, the forces, which operate on a flying aircraft, the weight, thrust, and aerodynamic forces, are all vector quantities. The resulting motion of the aircraft in terms of displacement, velocity, and acceleration are also vector quantities. These quantities can be determined by application of Newton's laws for vectors. The scalar quantities include most of the thermodynamic state variables involved with the propulsion system, such as the density, pressure, and temperature of the propellants. The energy, work, and entropy associated with the engines are also scalar quantities. There are some quantities, like speed, which have very special definitions for scientists. By definition, speed is the scalar magnitude of a velocity vector. A car going down the road has a speed of 50 mph. Its velocity is 50 mph in the northeast direction. It can get very confusing when the terms are used interchangeably! While Newton's laws describe the resulting motion of a solid, there are special equations which describe the motion of fluids, gases and liquids, through the propulsion system. For any physical system, the mass,...

...SCALAR Quantities
Scalar is the measurement of a medium strictly in magnitude. It is a physical quantity that is unchanged by coordinate system rotations or reflections (in Newtonian mechanics), or by Lorentz transformations or space-time translations (in relativity).
A scalar is a quantity which can be described by a single number, unlike vectors, tensors, etc. which are described by several numbers which describe magnitude and direction. A related concept is a pseudoscalar, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations. The concept of a scalar in physics is essentially the same as in mathematics.
An example of a scalar quantity is temperature: the temperature at a given point is a single number. Velocity, on the other hand, is a vector quantity: velocity in three-dimensional space is specified by three values; in a Cartesian coordinate system the values are the speeds relative to each coordinate axis.
VECTOR QUANTITIES
Vector is a measurement that refers to both the magnitude of the medium as well as the direction of the movement the medium has taken. It is is a quantity that has both magnitude and direction. It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity’s magnitude. Although a...

...Velocity is a vector physical quantity; both magnitude and direction are required to define it.
the length of an imaginary straight path, typically distinct from the path actually travelled by P.
Distance is a numerical description of how far apart objects are. In physics or everyday usage, distance may refer to a physical length, or an estimation
Time in physics is
defined by its measurement
Formula
Acceleration= change in velocity / time interval
Average velocity= displacement/time
S= (square root of) xsquared plus y squared
D=xt
X=d/t
Units
The gal, sometimes called galileo, (symbol Gal)
The SI unit of speed and velocity is the ratio of two — the meterper second.
The standard unit of displacement in the International System of Units ( SI ) is the meter (m).
Meter (m)
In the International System of Units (SI), the unit of time is the second (symbol:
Examples
At an amusement park, when you ride a rollercoaster is starts slowly, then faster, then slows down again.
If you walk to a campsite 1 mile away and then back to your start point within 1 hour:
Your average speed would be: total distance/time = 2 miles/1 hour or 2 miles per hour
How far does the earth travel in one year? In terms of distance, quite far (the circumference of the earth's orbit is nearly one trillion
meters), but in terms of displacement, not far at all (zero, actually). At the end of a year's time the earth is right back where it started from. It hasn't gone...

...HL Vectors Notes
1.
Vector or Scalar
Many physical quantities such as area, length, mass and temperature are completely described once the magnitude of the quantity is given. Such quantities are called “scalars.” Other quantities possess the properties of magnitude and direction. A quantity of this kind is called a “vector” quantity. Winds are usually described by giving their speed and direction; say 20 km/h north east. The wind speed and wind direction together form a vector quantity called the wind velocity. A force, for example, is characterized by its magnitude and direction of action. The force would not be completely specified by one of these properties without the other. The velocity of a moving body is determined by its speed and direction of motion. Acceleration and displacement are other examples of vector quantities.
A scalar is a quantity that has magnitude or size but no direction.
A vector is a quantity that has both magnitude and direction.
Displacement Vectors and Notation
[pic]
Vectors can be represented geometrically by arrows in 2- or 3-space; the direction of the arrow specifies the direction of the vector and the length of the arrow describes its magnitude. The first point in the arrow is called the initial point of the vector and the tip is...

...Calculus in 3D Geometry, Vectors, and Multivariate Calculus Zbigniew H. Nitecki
Tufts University
August 19, 2012
ii
This work is subject to copyright. It may be copied for non-commercial purposes.
Preface
The present volume is a sequel to my earlier book, Calculus Deconstructed: A Second Course in First-Year Calculus, published by the Mathematical Association in 2009. I have used versions of this pair of books for severel years in the Honors Calculus course at Tufts, a two-semester “boot camp” intended for mathematically inclined freshmen who have been exposed to calculus in high school. The ﬁrst semester of this course, using the earlier book, covers single-variable calculus, while the second semester, using the present text, covers multivariate calculus. However, the present book is designed to be able to stand alone as a text in multivariate calculus. The treatment here continues the basic stance of its predecessor, combining hands-on drill in techniques of calculation with rigorous mathematical arguments. However, there are some diﬀerences in emphasis. On one hand, the present text assumes a higher level of mathematical sophistication on the part of the reader: there is no explicit guidance in the rhetorical practices of mathematicians, and the theorem-proof format is followed a little more brusquely than before. On the other hand, the material being developed here is unfamiliar territory, for the intended audience, to a far greater degree...

...MIchael Driesen
Mrs. Rozell
Math 10H
17 December 2011
Vectors
Math is everywhere. No matter which way you look at it, it’s there. It is especially present in science. Most people don’t notice it, they have to look closer to find out what it is really made of. A component in math that is very prominent in science is the vector. What is a vector? A vector is a geometric object that has both a magnitude and a direction. A good example of a vector is wind. 30 MPH north. It has both magnitude,(in this case speed) and direction. Vectors have specific properties that make them very useful in real life applications. Through the use of these special objects, many advancements in the fields of math and science are available.
Representations
Vectors can sometimes be hidden behind basic objects. They are usually represented with an arrow on top of its starting point and terminal point, as shown here: The most common form of vector is the bound vector. All that means is that the starting point of the vector is the origin, or (0,0). The bound vector goes from the origin to it’s terminal point, which in this case can be (3,4). An easy way to write this
is A = (3,4), where A is the vector. On a graph, it looks like this:
That applies for two-dimensional vectors. Three dimensional...

...
1a. h=-4.9t^2+450
1b. h(t)=-4.9t^2+450
(h(2)-h(0))/(2-0)
((-4.9(〖2)〗^2+450)-(-4.9(0)^2+450))/2
=(430.4-450)/2
=-19.6
∴The average velocity for the first two seconds was 19.6 metres per second.
c. i)
i)
=
=-24.5
∴ The average velocity from is 24.5 metres/s.
ii)
= -14.7
iii)
= -12.25
∴ The average velocity from is 12.25 metres/s.
d) Instantaneous velocity at 1s:
=-9.8
∴ The instantaneous velocity at 1s is 9.8 metres/s.
2a)
=
=
=
=
=
b)
=
∴ The average rate of change from is -0.4g/s.
C)
∴The instantaneous rate at t = 2 seconds is -1.6g/s
3)
b)
=
=
=22
∴ from seconds the car moves at an average of 22m/s
c)
t=4
=
=16
∴ The instantaneous rate at 4s is 16m/s
4a) In order to determine the instantaneous rate of change of a function using the methods discussed in this lesson, we would use the formula where h will approach 0, and the closer it gets to 0 the more accurate our answer will be.
4b)
∴
=1
Therefore, = 1
5a)
Therefore the instantaneous rate at x=2 is 0.
5b)
Therefore at t=4 the instantaneous rate is 0 and the particle is at rest.
6a)
Rate of change is positive when:
Rate of change in negative when:
6b)
Rate of change is 0 when:
X=-1, x=1
6c)
Local Maximum: (-1,2)
Local Minimum: (1,-2)...

...Mehran University College
Of Engineering & Technology,
Khairpur Mir’s
VECTOR GROUPS
ENGR. AHSANULLAH MEMON
LECTURER
DEPARTMENT OF ELECTRICAL ENGINEERING MUCET KHAIRPUR MIRS
ZIGZAG CONNECTION OF TRANSFORMER
The zigzag connection of tranformer is also called the
interconnected star connection.
This connection has some of the features of the Y and
the ∆ connections, combining the advantages of both.
The zigzag transformer contains six coils on three
cores.
Its applications are for the deviation of a neutral
connection from an ungrounded 3-phase system and
the grounding of that neutral to an earth reference
point and harmonics mitigation.
It can cancel triplet (3rd, 9th, 15th, 21st, etc.)
harmonic currents.
INTRODUCTION
Secondary voltage waveforms are in phase
with the primary waveforms.
When two transformers are connected in
parallel, their phase shifts must be identical; if
not, a short circuit will occur when the
transformers are energized.”
When two transformers are connected in
parallel, their phase shifts must be identical; if
not, a short circuit will occur when the
transformers are energized.”
Vector Group of Transformer
The three phase transformer windings can be connected several
ways. Based on the windings’ connection, the vector group of
the transformer is determined.
The transformer vector group is indicated on the Name Plate of
transformer by the manufacturer.
The...

...The outcome at any stage depends only on the outcome of the previous stage. (c.) The probabilities are constant over time. If x0 is a vector which represents the initial state of a system, then there is a matrix M such that the state of the system after one iteration is given by the vector M x0 . Thus we get a chain of state vectors: x0 , M x0 , M 2 x0 , . . . where the state of the system after n iterations is given by M n x0 . Such a chain is called a Markov chain and the matrix M is called a transition matrix. The state vectors can be of one of two types: an absolute vector or a probability vector. An absolute vector is a vector whose entries give the actual number of objects in a give state, as in the ﬁrst example. A probability vector is a vector where the entries give the percentage (or probability) of objects in a given state. We will take all of our state vectors to be probability vectors from now on. Note that the entries of a probability vector add up to 1.
1
Note b =
Theorem 3. Let M be the transition matrix of a Markov process such that M k has only positive entries for some k. Then there exists a unique probability vector xs such that M xs = xs . Moreover limk→∞ M k x0 = xs for any initial state probability vector x0 . The vector xs...

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