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7.4 Dot Product of Algebraic Vectors A Dot Product for Standard Unit Vectors The dot product of the standard unit vectors is given by: r r r r r r i ⋅ i =1 j ⋅ j =1 k ⋅ k =1 r r r r r r i ⋅ j =0 j ⋅k =0 k ⋅i = 0 B Dot Product for two Algebraic Vectors The dot product of two algebraic vectors r r r r a = (a x , a y , a z ) = a x i + a y j + a z k and r r r r b = (b x , b y , b z ) = b x i + b y j + b z k is given by: r r a ⋅ b = a x bx + a y b y + a z bz Proof: r r r r r r r r a ⋅ b = (a x i + a y j + a z k ) ⋅ (bx i + b y j + bz k ) r r r r r r = (a x bx )(i ⋅ i ) + (a x b y (i ⋅ j ) + (a x bz )(i ⋅ k ) + r r r r r r + (a y bx )( j ⋅ i ) + (a y b y ( j ⋅ j ) + (a y bz )( j ⋅ k ) + r r r r r r + (a z bx )(k ⋅ i ) + (a z b y (k ⋅ j ) + (a z bz )(k ⋅ k )

= a x bx + a y b y + a z bz

Proof: r r r r i ⋅ i =|| i || || i || cos 0° = (1)(1)(1) = 1 r r r r i ⋅ j =|| i || || j || cos 90° = (1)(1)(0) = 0

r Ex 1. For each case, find the dot product of the vectors a r and b . r r a) a = (1,−2,0) , b = (0,−1,2) r r a ⋅ b = (1)(0) + (−2)(−1) + (0)(2) = 2 r r r r r r r b) a = −i + 2 j , b = i − 2 j − k r r a ⋅ b = (−1)(1) + (2)(−2) + (0)(−1) = −1 − 4 = −5 r r r r r c) a = (−1,1,−1) , b = −i + 2 j − 2k r r a ⋅ b = (−1)(−1) + (1)(2) + (−1)(−2) = 1 + 2 + 2 = 5

C Angle between two Vectors r r r The angle θ = ∠(a , b ) between two vectors a and r b (when positioned tail to tail) is given by: r r a x bx + a y b y + a z bz a ⋅b cosθ = r r = | a || b | a x 2 + a y 2 + a z 2 bx 2 + b y 2 + bz 2 Notes: r r 1. If cosθ = 1 then a ↑↑ b (vectors are parallel and have same direction). r r 2. If cosθ = −1 then a ↑↓ b (vectors are parallel but have opposite direction). r r 3. If cosθ = 0 then a ⊥ b (vectors are perpendicular to each other or orthogonal).

r Ex 2. For each case, find the angle between the vectors a r and b . r r a) a = (1,−2,−1)…...

...Excerpt More information 1 Vector and tensor analysis Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector ®elds that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely speci®ed, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete speci®cation. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without aecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magnitude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A) and use ordinary italic letters (such as A) for their magnitudes; in writing, vectors are usually ~ ~ represented by a letter with an arrow above it such as A. A given vector A (or A) can be......

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...Unit 1 Discussion 2 1. What main argument (or claim) does the author make? What contradictions (if any) does the author make in her argument (or claim)? What competing claims (points of view other than the main one the author defends) appear in the essay? The main point the author makes is to not just pass students through school, make them earn it. The contradiction to her argument is that people blame outside influences, such as drugs and divorce for those in high school not academics seriously. I notice a theme of needed to hold people accountable. 2. What viewpoints may audience members have that oppose the author's main argument? What evidence is there that the author anticipates these opposing viewpoints that her audience members might hold? How does the author defend her argument against those opposing viewpoints? Young adults going through school should not have it held against them for outside influences that affect their academics. The author does anticipate these viewpoints and addresses them. She defends it by showing that after high school, the world is waiting for you, and that it does not cut you any slack because you run into issues that could affect your performance. 3. What assumptions does the writer's title carry? They are a teacher and should be well versed in the challenges facing students. Being a night school teacher also gives the assumption that they are very familiar with the aftermath for the student that is just passed along in......

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... #2 The Study of Concurrent Forces with the Force Table Apparatus: Force table with 4 pulleys, centering ring and string, 50 g weight hangers, slotted weights, protractors, and rulers. Discussion: The force table is designed to help you study the properties of forces at known angles. Only when forces are along the same line do they add by ordinary algebra. If two or more forces on the same body form angles with each other, it is necessary to use geometry to find the amount and direction of their combined effect. Prior to Lab: Complete the calculations in the following. The component method of adding vectors is given here for three sample forces as follows: A = 2.45 N @ 40o B = 3.92 N @ 165o C = 3.43 N @ 330o Overview of the component method of vector addition. First make a neat drawing, not necessarily to exact scale, but reasonably accurate as to sizes and angles, placing the three forces on a diagram with a pair of x and y axes. Find the angle of each force with the x-axis. This angle is called the reference angle, and is the one used to calculate sines and cosines. Next compute the x- and y-components of the three forces, placing like components in columns. Place plus or minus signs on the various quantities according to whether an x-component is to the right or left of the origin, or whether a y-component is up or down relative to the origin. Add the columns with regard to sign (subtracting the minus quantities), and place the correct sign on each sum. The resulting...

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...UNIT 6: BUSINESS DECISION MAKING Unit 6: Unit code: QCF level: Credit value: Aim Business Decision Making D/601/0578 5 15 credits The aim of this unit is to give learners the opportunity to develop techniques for data gathering and storage, an understanding of the tools available to create and present useful information, in order to make business decisions Unit abstract In business, good decision making requires the effective use of information. This unit gives learners the opportunity to examine a variety of sources and develop techniques in relation to four aspects of information: data gathering, data storage, and the tools available to create and present useful information. ICT is used in business to carry out much of this work and an appreciation and use of appropriate ICT software is central to completion of this unit. Specifically, learners will use spreadsheets and other software for data analysis and the preparation of information. The use of spreadsheets to manipulate of numbers, and understanding how to apply the results, are seen as more important than the mathematical derivation of formulae used. Learners will gain an appreciation of information systems currently used at all levels in an organisation as aids to decision making. Learning outcomes On successful completion of this unit a learner will: 1 2 3 4 Be able to use a variety of sources for the collection of data, both primary and secondary Understand a range of techniques to analyse data effectively for...

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... 9 a. Computing Eigenvalues 9 b. Computing Eigen Vectors 10 5. Applications 10 a. Geology and Glaciology 10-11 b. Vibration Analysis 11-12 c. Tensor of Moment of Inertia 12 d. Stress Tensor 12 e. Basic Reproduction Number. 12 6. Conclusion 13 7. References 13 3 Abstract In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is complex numbers); the set of Cartesian the continuous functions, the multiplication is replaced by any the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as " This paper is about the various calculations various civil engineering problems like vibrational analysis,......

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...6302 Vector Aeromotive Corporation Vector sold exotic sports cars and was the only US based manufacturer. Their major competition, Ferrari and Lamborghini, took up 75% of the market share. Gerry’s idea was to make a car based off of aerospace technology. They created the V8 twin turbo which was highly advanced and priced. After selling a total of 13 cars, 45 people were employed. Vector built two other models to increase sell volume and decrease losses. Vector’s Board of Directors composed of 3 individuals, Berry with a background in real estate; John, a financial consultant, and Vector’s CFO; and Gerry, the president of Vector. Berry and John grew to disagree with Gerry more often; therefore, Gerry brought in Dan, an attorney and associate of Gerry, to help shift the balance in his favor. In June 1990, vector hired a vice president; this V.P did not agree with Gerry’s management style and reported it to the board. Dan Harnett even began to see the malevolent behaviors of Gerry, and called for his removal. The other board members did not agree to this because there was no one to take his place and Gerry was allowed to be a bad manager, according to his employment contract, he just could not do anything illegal. Instead the board decided to require Gerry to submit formal expense reports in order to get reimbursed for expenses. Dan later resigned from the board and was replaced by George, another associate of Gerry. June 1992, 2 million dollars was given to Vector......

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...VECTOR FUNCTIONS VECTOR FUNCTIONS Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal vectors. MOTION IN SPACE: VELOCITY AND ACCELERATION Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study: The motion of an object, including its velocity and acceleration, along a space curve. VELOCITY AND ACCELERATION In particular, we follow in the footsteps of Newton by using these methods to derive Kepler’s First Law of planetary motion. VELOCITY Suppose a particle moves through space so that its position vector at time t is r(t). VELOCITY Vector 1 Notice from the figure that, for small values of h, the vector r(t h) r(t ) h approximates the direction of the particle moving along the curve r(t). VELOCITY Its magnitude measures the size of the displacement vector per unit time. VELOCITY The vector 1 gives the average velocity over a time interval of length h. VELOCITY VECTOR Equation 2 Its limit is the velocity vector v(t) at time t : r(t h) r(t ) v(t ) lim h 0 h r '(t ) VELOCITY VECTOR Thus, the velocity vector is also the tangent vector and points in the direction of the tangent line. SPEED The speed of the particle at time t is the magnitude of the velocity vector, that is, |v(t)|. SPEED This is appropriate because, from Equation......

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...Tutorial 1 – Vector Calculus 1. Find the magnitude of the vector PQ with P (−1,2) and Q (5,5) . 2. Find the length of the vector v = 2,3,−7 . 3. Given the points in 3-dimensional space, P ( 2,1,5), Q (3,5,7), R (1,−3,−2) and S ( 2,1,0) . Does r PQ = RS ? ˆ ˆ 4. Find a vector of magnitude 5 in the direction of v = 3i + 5 ˆ − 2k . j r r ˆ ˆ ˆ j ˆ 5. Given u = 3i − ˆ − 6k and v = −i + 12k , find (a) u • v , r r (b) the angle between vectors u and v , r (c) the vector proju v , r r r r (d) the scalar component of v in the direction of u . 6. Given P (1,−1,3), Q ( 2,0,1) and R (0,2,−1) , find (a) the area of the triangle determined by the points P, Q and R. (b) the unit vector perpendicular to the plane PQR. 7. Find the volume of the parallelepiped determined by the vectors u = 4,1,0 , v = 2,−2,3 and r r r r r w = 0,2,5 . 8. Find the area of the parallelogram whose vertices are given by the points A (0, 0, 0), B (3, 2, 4), C (5, 1, 4) and D (2, -1, 0). ˆ j 9. Find the equation of the line through (2, 1, 0) and perpendicular to both i + ˆ and ˆ + k . j ˆ 10. Find the parametric equation of the line through the point (1, 0, 6) and perpendicular to the plane x+3y+z=5. 11. Determine whether the given lines are skew, parallel or intersecting. If the lines are intersecting, what is the angle between them? L1: x −1 y −3 z−2 = = 2 2 −1 x−2 y−6 z+3 L2 : = = 1 −1 3 12. Find the point in which the line x = 1 –t, y = 3t, z = 1 + t meets...

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...5 6 1 5 2 0 <= Available Input a text file that includes the number of processes, resources, and the matrixes for allocations, max, and available. Output System Safe or Unsafe 1) Read the # of processes and the # of resources 2) Read allocation, max and available for each process and each resource 3) Print whether this system is safe or not. Data Structures for the Banker’s Algorithm Let n = number of processes, and m = number of resources types. Available: Vector of length m. If available [j] = k, there are k instances of resource type Rj available Input.txt Input.txt Max: n x m matrix. If Max [i,j] = k, then process Pi may request at most k instances of resource type Rj Allocation: n x m matrix. If Allocation[i,j] = k then Pi is currently allocated k instances of Rj Need: n x m matrix. If Need[i,j] = k, then Pi may need k more instances of Rj to complete its task Need [i,j] = Max[i,j] – Allocation [i,j] Safety Algorithm 1. Let Work and Finish be vectors of length m and n, respectively. Initialize: Work = Available Finish [i] = false for i = 0, 1, …, n- 1 2. Find an i such that both: (a) Finish [i] = false (b) Needi ≤ Work If no such i exists, go to step 4 3. Work = Work + Allocationi Finish[i] = true go to step 2 4. If Finish [i] == true for all i, then the system is in a safe state...

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...Find parametric equations for the line L which contains A(1, 2, 3) and B(4, 6, 5). Solution: To get the parametric equations of L you need a point through which the line passes and a vector parallel to the line. −→ Take the point to be A and the vector to be the AB. The vector equation of L is −→ −→ r(t) = OA + t AB = 1, 2, 3 + t 3, 4, 2 = 1 + 3t, 2 + 4t, 3 + 2t , where O is the origin. The parametric equations are: x = 1 + 3t y = 2 + 4t, z = 3 + 2t t ∈ R. Problem 1(b) - Fall 2008 Find parametric equations for the line L of intersection of the planes x − 2y + z = 10 and 2x + y − z = 0. Solution: The vector part v of the line L of intersection is orthogonal to the normal vectors 1, −2, 1 and 2, 1, −1 . Hence v can be taken to be: i j k v = 1, −2, 1 × 2, 1, −1 = 1 −2 1 = 1i + 3j + 5k. 2 1 −1 Choose P ∈ L so the z-coordinate of P is zero. Setting z = 0, we obtain: x − 2y = 10 2x + y = 0. Solving, we ﬁnd that x = 2 and y = −4. Hence, P = 2, −4, 0 lies on the line L. The parametric equations are: x =2+t y = −4 + 3t z = 0 + 5t = 5t. Problem 2(a) - Fall 2008 Find an equation of the plane which contains the points P(−1, 0, 1), Q(1, −2, 1) and R(2, 0, −1). Solution: Method 1 −→ −→ Consider the vectors PQ = 2, −2, 0 and PR = 3, 0, −2 which lie parallel to the plane. Then consider the normal vector: i j k −→ −→ n = PQ × PR = 2 −2 0 3 0 −2 = 4i + 4j + 6k. So the equation of the plane is given by: 4, 4, 6 · x + 1, y , z − 1 = 4(x + 1) + 4y + 6(z − 1) = 0. Problem 2(a) - Fall......

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...Statement: Hills College’s Department of Residence life strives to provide students with a safe and comfortable living environment, where one can achieve academic, social, and personal growth in community settings. The Department of Residence Life expectations for students align with Hill’s College mission of forming the well-rounded student. 3. The overall mission of Hills College’s Department of Residence life is supported by Chickering’s 7 Vectors of Development and Chickerering’s 7 Vectors of Development illustrates how students develop in the college setting and how this development can affect him or her emotionally, socially, physically and intellectually in a college environment. The first 3 vectors are usually covered in the first years of college. Residence Life hopes to help guide students through as many vectors as possible. The Vectors are identified as developmental tasks for student’s to experience in order to become a complete individual. The vectors that Residence Life strives to help students accomplish are: * Vector 1: Developing Competence. Intellectual, Physical, and Interpersonal Competence are the 3 main area. Intellectual Competence could be attained with help from residence life by forming study groups or tutors in the hall. Study areas in residence halls is a definite plus. Students can interact and also help one another on subjects they are knowledgeable in. There are many things a residence life staff can do to help students develop......

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...International Journal of Engineering Trends and Technology- Volume3Issue3- 2012 Hiding Messages Using Motion Vector Technique In Video Steganography P.Paulpandi1, Dr.T.Meyyappan,M.sc.,M.Phil.,M.BA.,Ph.D2 Research Scholar1, Associate professor2 Department of Computer Science & Engineering, Alagappa University,Karaikudi. Tamil Nadu,India. Abstract- Steganography is the art of hiding information in ways that avert the revealing of hiding messages.Video files are generally a collection of images. so most of the presented techniques on images and audio can be applied to video files too. The great advantages of video are the large amount of data that can be hidden inside and the fact that it is a moving stream of image. In this paper, we proposed a new technique using the motion vector, to hide the data in the moving objects. Moreover, to enhance the security of the data, the data is encrypted by using the AES algorithm and then hided. The data is hided in the horizontal and the vertical components of the moving objects. The PSNR value is calculated so that the quality of the video after the data hiding is evaluated. Keywords- Data hiding, Video Steganography,PSNR, Moving objects, AES Algorithm. I. INTRODUCTION Since the rise of the Internet one of the most important factors of information technology and communication has been the security of information. Steganography is a technology that hides a user defined information within an object, a text...

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...exploit mainly the textual content of Web documents for retrieval. However, more recently Web search engines also start to exploit link information and even image information. The three tasks of a Web search engine for retrieval are: 1. extracting the textual features, which are the words or terms that occur in the documents. We assume that the web search engine has already collected the documents from the Web using a Web crawler. 2. support the formulation of textual queries. This is usually done by allowing the entry of keywords through Web forms. 3. computing the similarity of documents with the query and producing from that a ranked result. Here Web search engines use standard text retrieval methods, such as Boolean retrieval and vector space retrieval. We will introduce these methods in detail in this lecture later. 5 The heart of an information retrieval system is its retrieval model. The model is used to capture the meaning of documents and queries, and determine from that the relevance of documents with respect to queries. Although there exist a number of intuitive notions of what determines relevance one must keep clearly in mind that it is not an objective measure. The quality of a retrieval system can principally only be determined through the degree of satisfaction of its users. This is fundamentally different to database querying, where there exist criteria for correct query answering that can be formally verified, e.g., whether a result set retrieved......

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...Support Vector Machines Operations Management Project Report by- Suryansh Kapoor PGPM (2011 – 2013) 11P171 Section – ‘C’ Supervised by- Prof. Manoj Srivastava Abstract In today’s highly competitive world markets, high reliability plays increasingly important role in the modern manufacturing industry. Accurate reliability predictions enable companies to make informed decisions when choosing among competing designs or architecture proposals. This is all the more important in case of specialized fields where operations management is a necessary requirement. Therefore, predicting machine reliability is necessary in order to execute predictive maintenance, which has reported benefits include reduced downtime, lower maintenance costs, and reduction of unexpected catastrophic failures. Here, the role of Support Vector Machines or SVMs comes in to predict the reliability of the necessary equipment. SVMs are cited by various sources in the field of medical researches⁶ and other non-mining fields¹ to be better than other classifying methods like Monte-Carlo simulation etc. because SVM models have nonlinear mapping capabilities, and so can more easily capture reliability data patterns than can other models. The SVM model minimizes structural risk rather than minimizing training errors improves the generalization ability of the models. Contents 1. Objective 2. Literature Review * Introduction of Reliability *......

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