The document discusses vectors and trigonometry. It begins by defining vectors as quantities with both magnitude and direction, such as displacement and velocity. It then covers geometric descriptions of vectors using arrows to represent direction. It also discusses vector operations like addition and scalar multiplication analytically and geometrically. Finally, it shows how vectors can model velocity and how to calculate the true velocity of an object when it experiences additional velocities like wind.
As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two vectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors defined for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Using a Common Theme to Find Intersections of Spheres with Lines and Planes v...James Smith
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.
As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two vectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors defined for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Using a Common Theme to Find Intersections of Spheres with Lines and Planes v...James Smith
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A Very Brief Introduction to Reflections in 2D Geometric Algebra, and their U...James Smith
This document discusses reflections of vectors and of geometrical products of two vectors, in two-dimensional Geometric Algebra (GA). It then uses reflections to solve a simple tangency problem.
Via Geometric (Clifford) Algebra: Equation for Line of Intersection of Two Pl...James Smith
As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A Very Brief Introduction to Reflections in 2D Geometric Algebra, and their U...James Smith
This document discusses reflections of vectors and of geometrical products of two vectors, in two-dimensional Geometric Algebra (GA). It then uses reflections to solve a simple tangency problem.
Via Geometric (Clifford) Algebra: Equation for Line of Intersection of Two Pl...James Smith
As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
Chapter 12
Section 12.1: Three-Dimensional Coordinate Systems
We locate a point on a number line as one coordinate, in the plane as an ordered pair, and in
space as an ordered triple. So we call number line as one dimensional, plane as two
dimensional, and space as three dimensional co – ordinate system.
In three dimensional, there is origin (0, 0, 0) and there are three axes – x -, y - , and z – axis. X –
and y – axes are horizontal and z – axis is vertical. These three axes divide the space into eight
equal parts, called the octants. In addition, these three axes divide the space into three
coordinate planes.
– The xy-plane contains the x- and y-axes. The equation is z = 0.
– The yz-plane contains the y- and z-axes. The equation is x = 0.
– The xz-plane contains the x- and z-axes. The equation is y = 0.
If P is any point in space, let:
– a be the (directed) distance from the yz-plane to P.
– b be the distance from the xz-plane to P.
– c be the distance from the xy-plane to P.
Then the point P by the ordered triple of real numbers (a, b, c), where a, b, and c are the
coordinates of P.
– a is the x-coordinate.
– b is the y-coordinate.
– c is the z-coordinate.
– Thus, to locate a point (a, b, c) in space, start from the origin (0, 0, 0) and move a
units along the x-axis. Then, move b units parallel to the y-axis. Finally, move c
units parallel to the z-axis.
The three dimensional Cartesian co – ordinate system follows the right hand rule.
Examples:
Plot the points (2,3,4), (2, -3, 4), (-2, -3, 4), (2, -3, -4), and (-2, -3, -4).
The Cartesian product x x = {(x, y, z) | x, y, z in } is the set of all ordered triples of
real numbers and is denoted by 3 .
Note:
1. In 2 – dimension, an equation in x and y represents a curve in the plane 2 . In 3 –
dimension, an equation in x, y, and z represents a surface in space 3 .
2. When we see an equation, we must understand from the context that it is a curve in the
plane or a surface in space. For example, y = 5 is a line in 2 �but it is a plane in 3 �
������
3. in space, if k, l, & m are constants, then
– x = k represents a plane parallel to the yz-plane ( a vertical plane).
– y = k is a plane parallel to the xz-plane ( a vertical plane).
– z = k is a plane parallel to the xy-plane ( a horizontal plane).
– x = k & y = l is a line.
– x = k & z = m is a line.
– y = l & z = m is a line.
– x = k, y = l and z = m is a point.
Examples: Describe and sketch y = x in 3
Example:
Solve:
Which of the points P(6, 2, 3), Q(-5, -1, 4), and R(0, 3, 8) is closest to the xz – plane? Which point
lies in the yz – plane?
Distance between two points in space:
We simply extend the formula from 2 to . 3 . The distance |p1 p2 | between the points
P1(x1,y1, z1) and P2(x2, y2, z2) is: 2 2 21 2 2 1 ...
Three Solutions of the LLP Limiting Case of the Problem of Apollonius via Geo...James Smith
This document adds to the collection of solved problems presented in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016,http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra, http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra, and http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the LLP limiting case of the Problem of Apollonius in two ways.
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2. 2
INFORMATION - TUESDAY 1 AUGUST 2017
1. CLASS TEST 11. CLASS TEST 1 on par. 6.5 is scheduled foron par. 6.5 is scheduled for
Thursday 3 August.Thursday 3 August.
2. Worksheet 22. Worksheet 2 on par. 6.3 (vectors) & par. 6.5
(trigonometry form of complex numbers) is
scheduled for Tuesday 8 August. You
can use your textbook and class notes.
3. Find all info on uLink.
5. 5
Vectors in Two Dimensions
In applications of mathematics, certain quantities
are determined completely by their magnitude;
length, mass, area, temperature, and energy.
We speak of a length of 5 m or a mass of 3 kg;
only one number is needed to describe each of
these quantities. Such a quantity is called a
scalarscalar.
On the other hand, to describe the displacement
of an object, two numbers are required: the
magnitudemagnitude and the directiondirection of the
displacement.
6. 6
Vectors in Two Dimensions
To describe the velocity of a moving object, we
must specify both the speedspeed and the directiondirection
of travel.
Quantities such as displacement, velocity,
acceleration, and force that involve magnitude
as well as direction are called directed
quantities.
One way to represent such quantities
mathematically is through the use of vectorsvectors.
8. 8
Geometric Description of Vectors
A vector in the plane is a line segment with an assigned
direction. We sketch a vector as shown in Figure 1 with an
arrow to specify the direction.
We denote this vector by . Point A
is the initial point, and B is the terminal
point of the vector .
The length of the line segment AB is called
the magnitude or length of the vector and
is denoted by
Figure 1
9. 9
Geometric Description of Vectors
We use boldface letters to denote vectors. Thus we write
u = . Two vectors are considered equal if they have
equal magnitude and the same direction. Thus all the
vectors in Figure 2 are equal.
Figure 2
10. 10
Geometric Description of Vectors
This definition of equality makes sense if we think of a
vector as representing a displacement.
Two such displacements are the same if they have
equal magnitudes and the same direction.
So the vectors in Figure 2 can be thought of as the
same displacement applied to objects in different
locations in the plane.
11. 11
Geometric Description of Vectors
If the displacement u = is followed by the displacement
v = , then the resulting displacement is as shown in
Figure 3.
Figure 3
12. 12
Geometric Description of Vectors
In other words, the single displacement represented by the
vector has the same effect as the other two
displacements together. We call the vector the sumsum of
the vectors and , and we write .
(The zero vector, denoted by 0, represents no
displacement.)
Thus to find the sum of any two vectors
u and v, we sketch vectors equal to
u and v with the initial point of one at the
terminal point of the other
(see Figure 4(a)).
Addition of vectors
Figure 4(a)
13. 13
Geometric Description of Vectors
If we draw u and v starting at the same point, then u + v is
the vector that is the diagonal of the parallelogram formed
by u and v shown in Figure 4(b).
Figure 4(b)
Addition of vectors
14. 14
Geometric Description of Vectors
If a is a real number and v is a vector, we define a
new vector av as follows: The vector av has
magnitude |a||v| and has the same direction as v if
a > 0 and the opposite direction if a < 0. If a = 0, then
av = 0, the zero vector.
This process is called multiplication of a vector bymultiplication of a vector by
a scalara scalar. Multiplying a vector av by a scalar has the
effect of stretching or shrinking the vector.
15. 15
Geometric Description of Vectors
Figure 5 shows graphs of the vector av for
different values of a. We write the vector (–1) v as
–v. Thus –v is the vector with the same length as v
but with the opposite direction.
Figure 5
16. 16
Geometric Description of Vectors
The differencedifference of two vectors u and v is defined by
u – v = u + (–v). Figure 6 shows that the vector
u – v is the other diagonal of the parallelogram formed
by u and v.
Figure 6
18. 18
Vectors in the Coordinate Plane
In Figure 7, to go from the initial point of the vector v
to the terminal point, we move a units to the right and
b units upward. We represent v as an ordered pair of
real numbers.
v = a, b
where a is the horizontal component of v and b is
the vertical component of v.
Figure 7
19. 19
Vectors in the Coordinate Plane
Using Figure 8, we can state the relationship
between a geometric representation of a vector and
the analytic one as follows.
Figure 8
21. 21
Example 1 – Describing Vectors in Component Form
(a) Find the component form of the vector u with
initial point (–2, 5) and terminal point (3, 7).
(b) If the vector v = 3, 7 is sketched with initial
point (2, 4), what is its terminal point?
(c) Sketch representations of the vector w = 2, 3
with initial points at (0, 0), (2, 2), (–2, –1) and
(1, 4).
22. 22
Example 1 – Solution
(a) The desired vector is
u = 3 – (–2), 7 – 5 = 5, 2
(b) Let the terminal point of v be (x, y). Then
x – 2, y – 4 = 3, 7
So x – 2 = 3 and y – 4 = 7, or x = 5 and y = 11. The
terminal point is (5, 11).
23. 23
Example 1 – Solution
(c) Representations of the vector w are sketched in
Figure 9.
cont’d
Figure 9
24. 24
Vectors in the Coordinate Plane
We now give analytic definitions of the various
operations on vectors that we have described
geometrically.
Two vectors are equal if they have equal magnitude
and the same direction.
For the vectors u = a1, b1 and v = a2, b2, this
means that a1 = a2 and b1 = b2. In other words, two
vectors are equalequal if and only if their corresponding
components are equal. Thus all the arrows in Figure
9 represent the same vector.
25. 25
Vectors in the Coordinate Plane
Applying the Pythagorean Theorem to the triangle in
Figure 10, we obtain the following formula for the
magnitude of a vector.
Figure 10
26. 26
Vectors in the Coordinate Plane
The following definitions of addition, subtraction, and scalar
multiplication of vectors correspond to the geometric
descriptions given earlier.
Figure 11 shows how the analytic
definition of addition corresponds
to the geometric one.
Figure 11
27. 27
Example 2 – Operations with Vectors
If u = 2, –3 and v = –1, 2, find u + v, u – v, 2u, –3v, and
2u + 3v.
Solution:
By the definitions of the vector operations we have
u + v = 2, –3 + –1, 2 = 1, –1
u – v = 2, –3 – –1, 2 = 3, –5
2u = 22, –3 = 4, –6
29. 29
Vectors in the Coordinate Plane
The following properties for vector operations can be
easily proved from the definitions. The zero vectorzero vector is
the vector 0 = 0, 0. It plays the same role for addition
of vectors as the number 0 does for addition of real
numbers.
30. 30
Vectors in the Coordinate Plane
A vector of length 1 is called a unit vectorunit vector. For instance,
the vector w = is a unit vector. Two useful unit vectors
are i and j, defined by
i = 1, 0 j = 0, 1
Figure 12
31. 31
Vectors in the Coordinate Plane
These vectors are special because any vector can be
expressed in terms of them.
Figure 13
32. 32
Example 3 – Vectors in Terms of i and j
(a) Write the vector u = 5, –8 in terms of i and j.
(b) If u = 3i + 2j and v = –i + 6j, write 2u + 5v in terms of
i and j.
Solution:
(a) u = 5i + (–8)j = 5i – 8j
33. 33
Example 3 – Solution
(b) The properties of addition and scalar multiplication of
vectors show that we can manipulate vectors in the
same way as algebraic expressions.
Thus,
2u + 5v = 2(3i + 2j) + 5(–i + 6j)
= (6i + 4j) + (–5i + 30j)
= i + 34j
cont’d
34. 34
Vectors in the Coordinate Plane
Let v be a vector in the plane with its initial point at
the origin. The direction of v is , the smallest
positive angle in standard position formed by the
positive x-axis and v .
Figure 14
35. 35
Vectors in the Coordinate Plane
If we know the magnitude and direction of a vector,
then Figure 14 shows that we can find the horizontal
and vertical components of the vector.
36. 36
Example 4 – Components and Direction of a Vector
(a) A vector v has length 8 and direction /3. Find the
horizontal and vertical components, and write v in terms
of i and j.
(b) Find the direction of the vector u = – + j.
Solution:
(a) We have v = a, b, where the components are given by
and
Thus v = 4, =
37. 37
Example 4 – Solution
(b) From Figure 15 we see that the direction has the
property that
Thus the reference angle for is /6. Since the terminal
point of the vector u is in Quadrant II, it follows that
= 5/6.
cont’d
Figure 15
39. 39
Using Vectors to Model Velocity and Force
The velocityvelocity of a moving object is modeled by a
vector whose direction is the direction of motion and
whose magnitude is the speed.
Figure 16 shows some vectors u,
representing the velocity of wind
flowing in the direction N 30 E,
and a vector v, representing the
velocity of an airplane flying
through this wind at the point P.
Figure 16
40. 40
Using Vectors to Model Velocity and Force
It’s obvious from our experience that wind affects both
the speed and the direction of an airplane.
Figure 17 indicates that the true velocitytrue velocity of the plane
(relative to the ground) is given by the vector w = u + v.
Figure 17
41. 41
Example 5 – The True Speed and Direction of an Airplane
An airplane heads due north at 300 mi/h. It
experiences a 40 mi/h crosswind flowing in the
direction N 30 E, as shown in Figure 16.
Figure 16
42. 42
Example 5 – The True Speed and Direction of an Airplane
(a) Express the velocity v of the airplane relative
to the air, and the velocity u of the wind, in
component form.
(b) Find the true velocity of the airplane as a
vector.
(c) Find the true speed and direction of the
airplane.
43. 43
The velocity of the airplane relative to the air is
v = 0i + 300j = 300j. By the formulas for the components of
a vector, we find that the velocity of the wind is
u = (40 cos 60°)i + (40 sin 60°)j
20i + 34.64j
Example 5(a) – Solution
44. 44
Example 5(b) – Solution
The true velocity of the airplane is given by the vector
w = u + v:
w = u + v =
20i + 334.64j
cont’d
45. 45
Example 5(c) – Solution
The true speed of the airplane is given by the magnitude
of w:
|w| 335.2 mi/h
The direction of the airplane is the direction of the
vector w. The angle has the property that
tan 334.64/20 =16.732, so 86.6.
Thus the airplane is heading in the direction N 3.4 E.
cont’d