1. A ship sailing on the open sea leaves Port A for Port B at a bearing of N25oW. A wind of 6 km/h on a bearing of N10oE blows the ship off course. If the ship is capable of 35 km/h in still water, find the new speed and direction relative to the shore.

2. A boat is capable of 20 km/h in still water. You wish to cross the river to a point directly across from your present position. At what angle to the bank should you steer the boat if the current is 8 km/h?

3. A boat is capable of 20 km/h in still water. You wish to cross the river downstream so that the angle the boat's push makes with the bank is 50o. At what angle to the bank should the boat steer if the current is 8 km/h? How long will it take to cross if the river is 1.7 km wide?

4. A boat is capable of 20 km/h in still water. You wish to cross the river to a point 0.6 km upstream from your present position. If the current is 8 km/h and the river is 1.5 km wide, at what angle to the bank should you steer the boat? How long will it take you to cross?

5. An aircraft is currently on course flying from A to B, a distance of 400 km, on a bearing of S20oE at 350 km/h. A 50 km/h wind blowing S80oE starts to blow the aircraft off-course. At what new bearing should the pilot steer in order to stay on course? What will the new speed be after the course correction?

6. You wish to swim across a river that is 0.3 km wide to a point directly across from your present position. If you can swim at a constant speed of 4 km/h in still water and the current is 2.5 km/h, at what angle to the bank should you swim so that you end up directly across on the other side ( i.e. your actual path is 90o to the bank)? What is your speed relative to the bank?

7. Three coplanar forces of 10 N, 20 N and x N act on a body and maintain it in a state of equilibrium. The x force acts along the horizontal and the 10 N force at...

...3r21. ABCD is a rectangle and O is the midpoint of [AB].
Express each of the following vectors in terms of and
(a)
(b)
(c)
(Total 4 marks)
2. The vectors , are unit vectors along the x-axis and y-axis respectively.
The vectors = – + and = 3 + 5 are given.
(a) Find + 2 in terms of and .
A vector has the same direction as + 2 , and has a magnitude of 26.
(b) Find in terms of and .
(Total 4 marks)
3. The circle shown has centre O and radius 6. is the vector , is the vector and is the vector .
(a) Verify that A, B and C lie on the circle.
(3)
(b) Find the vector .
(2)
(c) Using an appropriate scalar product, or otherwise, find the cosine of angle .
(3)
(d) Find the area of triangle ABC, giving your answer in the form a , where a ∈ .
(4)
(Total 12 marks)
4. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).
(a) Find the vectors and .
(b) Find the angle between the diagonals of the quadrilateral OABC.
(Total 4 marks)
5. Find a vector equation of the line passing through (–1, 4) and (3, –1). Give your answer in the form r = p + td, where t ∈ R
(Total 4 marks)
6. In this question, the vector km represents a displacement due east, and the vector km a...

...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...

...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...

...ECE 352 VECTOR ANALYSIS
DEL OPERATOR
GROUP 3 Andaya, Rizalyn Ramos, Maria Issa P.
∇
Del is a symbol used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. Del may denote the gradient (locally steepest slope), the divergence of a vector field, or the curl (rotation) of a vector field. The symbol ∇ can be interpreted as avector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.
A. GRADIENT If a scalar function , , is continuously differentiable with respect to its variables x, y, z through the region, then the gradient of , written grad , is defined as the vector grad = Using the vector differential operator ∇, ∇= We can write the gradient of as
+
+
+
+
grad = ∇ = ( = Example: If =
+ +
+ +
)
, determine the grad at the point P(1,3,2).
By definition: grad = ∇ =
+
+
All we have to do then is to find the partial derivatives at x=1, y=3, z=2 and insert their values.
Since = = =
= = = = 2(1) (3) (2)3 + (3)2(2)2 (1)2(2)3 + 2(1) (3) (2)2 3(1) (3) (2)2 + 2(1) (3)2(2) = 84 = 32 = 72
grad
= ∇ = 84 i + 32 j +72 k
B. DIVERGENCE If a vector...

...1. Gradient of a scalar field function
Scalar Function:
Generally, What Is Scalar Function?
The Answer Is that a scalar function may be defined as A function of one or more variables whose range is one-dimensional, as compared to a vector function, whose range is three-dimensional (or, in general, -dimensional).
Scalar Field
When We Talk about Scalar Field, We Are Talking about the Scalar Function Being Applied to a Space (More like Euclenoid Space etc) or, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space (or space-time)
Gradient
Gradient of Scalar Field Function (E.g. Pressure, Temperature etc) will be the Vectors Which Would Eventually Point towards the Direction of Maximum Magnitude Increase.
Temperature Gradient (Gradient of Scalar function “Temperature”)
A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature gradient is a dimensional quantity expressed in units of degrees (on a particular temperature scale) per unit length. The SI unit is kelvin per meter (K/m).
The Application(s):
Weather and climate relevance...

...{draw:frame}
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I.INTRODUCTION
*II.*BASIC CONCEPT OF SVPWM
Space Vector Modulation treats the two level inverter of fig.1 as a single unit which can be driven to eight unique states that each state creates a corresponding voltage vector. An electric-motor control system, comprising:
*a two level voltage source inverter.
III.*IMPLE*MENTATION OF SPACE VECTOR PWM
For the three phase two-level PWM inverter as shown in Fig.1, the switch function is denoted by (1) ,shown below. where x=R,Y,B;"1"denotes Vdc/2 at the inverter output (a,b,c)with reference to point neutral;"0"denotes –Vdc/2;O is the neutral point of the dc bus.
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Fig. 1. Three phase two level inverter
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Fig.2 eight switching states
{draw:frame}
Fig. 3 Voltage vector space.
In the vector space, according to the equivalence principle the following conditions are obtained.
The following steps are carried out to implement SVPWM
Step.1 Determine the possible switching vectors in the output voltage space.
Step.3 Calculate the time variables
Step.4 switching time calculation to each sector
There are eight possible switching states for the inverter at any instant of time. The eight states of the inverter and the corresponding space- phasors of the output voltages are shown in fig 2 and 3 respectively. It can be observed that at any instant of time there...

...line that is perpendicular to the line with equation and that passes through the point with coordinates (2, 1). What is the perpendicular distance from the origin to the line with equation ?
3) Solve the inequality 2
4)Consider the vectors a = i − j + k, b = i + 2 j + 4k and c = 2i − 5 j − k.
(a)Given that c = ma + nb where m, n , find the value of m and of n.
(5)
(b)Find a unit vector, u, normal to both a and b.
(5)
(c)The plane 1 contains the point A (1, –1, 1) and is normal to b. The plane intersects the x, y and z axes at the points L, M and N respectively.
(i)Find a Cartesian equation of 1.
(ii)Write down the coordinates of L, M and N.
(5)
(d)The line through the origin, O, normal to π1 meets π1 at the point P.
(i)Find the coordinates of P.
(ii)Hence find the distance of π1 from the origin.
(7)
(e)The plane 2 has equation x + 2y + 4z = 4. Calculate the angle between 2 and a line parallel to a.
(5)
(Total 27 marks)
5)Two planes 1 and 2 are represented by the equations
1: r =
2: 2x – y – 2z = 4.
(a)(i)Find
(ii)Show that the equation of 1 can be written as x − 2y + 2z =11.
(4)
(b)Show that 1 is perpendicular to 2.
(4)
(c)The line l1 is the line of intersection of 1 and 2.
Find the vector equation of l1, giving the answer in parametric form.
(5)
(d)The line l2 is parallel to both 1 and 2, and passes through P(3, –5, –1).
Find an equation for l2 in Cartesian form.
(3)...