Scalar Products

10 Key excerpts on "Scalar Products"

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  eBook - ePub

  Vectors in Physics and Engineering

  • Alan Durrant(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  vector product because the product is itself another vector. Both kinds of product have wide applications in science and technology, as well as in mathematics. We begin with the scalar product.

  2.1 The Scalar Product

  The scalar product is associated with the idea of projection. An everyday example of projection is the shadow of a solid body cast on a flat surface by a source of light such as the Sun. This is an example of the projection of a solid body onto a plane. We are interested in the projection of one vector onto another vector. This is illustrated in Fig 2.1a which shows a vector a directed at an angle a from the direction of a vector b . The point P is obtained by dropping a perpendicular from the end point of a onto the line of b . The length OP = | a | cosα is the projection of vector a onto vector b . We often refer to this as the projection of a onto the direction of b or onto the unit vector

  b ^

  . Note that the projection of a onto b depends on the direction but not the magnitude of b . The projection is a positive number when the angle a is an acute angle as in Fig 2. 1a ; it can be zero or negative as illustrated in Figs 2.1b and c .

  Fig 2.1

  The length OP = |a | cosα is the projection of a into b ; it can be (a) postivie, (b) zero or (c) negative.

  If you have studied mechanics you may have come across projections in the context of the work done by a force. Fig 2.2 shows a force F applied to a body which undergoes a displacement s . The work done by the force is the projection of the force onto the displacement, |F | cosα, times the magnitude |s | of the displacement

  work done = 

  | F |

  | s |  cos α

  Fig 2.2

  The horse pulls the canal boat with a force F directed at an angle α to the displacement s = AB . The work done by the force when the boat moves from A to B is |F | |s

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  Fourier Methods in Imaging

  • Roger L. Easton Jr.(Author)
  • 2010(Publication Date)
  • Wiley

    (Publisher)

  In other words, this scalar product is a measure of the geometric projection of x onto ˆ a , as shown in Figure 3.4. If the reference vector is not normalized to unit length, then the linearity of the scalar product allows the result to be scaled by |a | −1 to obtain the correct-length projection of x in the direction of a : Projection of x onto a = a • x |a | (3.20) That the scalar product of any two vectors with θ = π/2 radians is zero is evident from its dependence on cos[θ ] in Equation (3.19); such vectors are said to be perpendicular or orthogonal. Two vectors that have unit length and are orthogonal are said to be orthonormal. Equation (3.19) also establishes that the scalar product of two vectors will be negative when θ lies within the semi-closed interval (+π/2, +π ], which means that the projection of x onto a points in the direction opposite to that of a . 3.2 Matrices 3.2.1 Simultaneous Evaluation of Multiple Scalar Products The concept of the scalar product of two vectors in Equation (3.9) may be extended to compute an ensemble of projections of a single N -D “input” vector x onto each of a set of M N -D unit-length “reference” vectors ˆ a m . For the moment, all components of all vectors still are assumed to be real Vectors with Real-Valued Components 35 Figure 3.4 Graphical interpretation of the projection of the 2-D vector x with real-valued components onto the unit vector ˆ a to produce |x | cos[θ ]. valued; the extension to complex-valued vectors will be discussed later. The ensemble of Scalar Products yields a set of M scalars that are distinguished by the index m assigned to each reference vector ˆ a m : ˆ a m • x = ( ˆ a m ) T x = N−1  n=0 ( ˆ a m ) n x n ≡ b m (3.21) Each of the M output scalars {b m } represents the projection of the N -D vector x onto a specific N -D “reference” vector ˆ a m , where the index m specifies the particular reference vector.

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  Essential Mathematics for Engineers and Scientists

  • Thomas J. Pence, Indrek S. Wichman(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  1.5 THE SCALAR PRODUCT Our development in Sections 1.2–1.4 required no concept of vector length or magnitude. Certainly the examples in R n and especially those in R 3 contain the notion of vector length, but this was never used in the development. Indeed, all of the analysis in Sections 1.2–1.4 was based on the algebraic notion of linear combination. The notion of linear combination also does not by itself give rise to a concept of vector orientation or direction, although we can say two vectors point in the same direction if they are positive multiples of each other, and point in diametrically opposite directions if they are negative multiples of each other. Barring these two special cases, linear combination by itself does not naturally generate a notion of how different in direction two vectors are from each other. Both magnitude and direction are useful concepts for vectors in general. To quantify these ideas in a general way it is useful to first introduce the notion of a scalar product of two vectors. The scalar product is often called the inner product, and in vector analysis is called the dot product. Whichever name it is given, this operation begins with two vectors and delivers a scalar. Given a vector space, there are a variety (actually, an infinite number) of possible Scalar Products, although one of them is used so frequently that it has become the de facto inner product. For the vector space R n this standard scalar product is the familiar x · y = ⎡ ⎢ ⎢ ⎢ ⎣ x 1 x 2 . . . x n ⎤ ⎥ ⎥ ⎥ ⎦ · ⎡ ⎢ ⎢ ⎢ ⎣ y 1 y 2 . . . y n ⎤ ⎥ ⎥ ⎥ ⎦ = x 1 y 1 + x 2 y 2 + · · · + x n y n , (1.22) which transforms R n × R n → R. Equation (1.22) is the standard example of the more general concept of a scalar product (·, ·): R n × R n → R. Given two vectors x and y from a vector space, their inner product is generally denoted by (x, y). Often we will work with the 22 Finite-Dimensional Vector Spaces vector space R n using the standard inner product defined by the formula (1.22).

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  A Student's Guide to Vectors and Tensors

  • Daniel A. Fleisch(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Vector operations If you were tracking the main ideas of Chapter 1 , you should realize that vectors are representations of physical quantities – they’re mathematical tools that help you visualize and describe a physical situation. In this chapter, you can read about a variety of ways to use those tools to solve problems. You’ve already seen how to add vectors and how to multiply vectors by a scalar (and why such operations are useful); this chapter contains many other “vector oper-ations” through which you can combine and manipulate vectors. Some of these operations are simple and some are more complex, but each will prove useful in solving problems in physics and engineering. The first section of this chapter explains the simplest form of vector multiplication: the scalar product. 2.1 Scalar product Why is it worth your time to understand the form of vector multiplication called the scalar or “dot” product? For one thing, forming the dot product between two vectors is very useful when you’re trying to find the projection of one vector onto another. And why might you want to do that? Well, you may be interested in knowing how much work is done by a force acting on an object. The first instinct of many students is to think of work as “force times distance” (which is a reasonable starting point). But if you’ve ever taken a course that went a bit deeper than the introductory level, you may remember that the definition of work as force times distance applies only to the special case in which the force points in exactly the same direction as the displacement of the object. In the more general case in which the force acts at some angle to the direction of the displacement, you have to find the component of the force along the displacement. That’s one example of exactly what the dot product can do for you, and you’ll find more in the problems at the end of this chapter. 25

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  What Every Engineer Should Know about MATLAB® and Simulink®

  • Adrian B. Biran(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Given two vectors, V 1 , V 2 , and knowing that the angle between them is α, their scalar , also called dot product , is defined as p = V1 · V 2 = V 1 · V 2 cos α (2.1) Vectors and matrices 67 The result is a scalar, hence the first name of the product. The dot, ‘ · ’, used to mark this product explains the other name. The operation is performed in MATLAB by the function dot . As an example let us define two vectors and calculate their dot product: V1 = [ 3.4641; 2 ]; V2 = [ 2.5; 4.3301 ]; p = dot(V1, V2) p = 17.3205 To verify that the result corresponds to the definition given above, calculate the lengths of the two vectors (using the function norm ) and the angle between them. norm(V1) ans = 4.0000 norm(V2) ans = 5.0000 beta = atand(V1(2)/V1(1)); gamma = atand(V2(2)/V2(1)); alpha = gamma -beta; p = norm(V1)*norm(V2)* cosd(alpha) We obtain the same result as the one calculated with the command dot . It is easy to prove that, if the vector V 1 has the components V 1 x , V 1 y , and the vector V 2 , the components V 2 x , V 2 y , then the product V 1 · V 2 equals V 1 x V 2 x + V 1 y V 2 y . Let us check this in MATLAB: p1 = V1(1)*V2(1) + V1(2)*V2(2) p1 = 17.3205 p -p1 ans = 0 The generalization to three and more dimensions is straightforward. The scalar product enables us to ‘measure’ lengths and angles. The length can be a displacement or the magnitude of a vector. For example consider the vector V 1 with the components defined above; its magnitude can be calculated as 68 What every engineer should know about MATLAB and Simulink V 1 = V 1 · V 1 = V 2 1 x + V 2 1 y With the values exemplified above, V1 = [ 3.4641; 2 ]; sqrt(dot(V1, V1)) ans = 4.0000 We recovered the same value as that obtained with the MATLAB built-in function norm . The use of the latter is simpler; however, it is important to show that the task can be carried on by calling the dot product. Let us show now how the dot product enables us to ‘measure’ angles.

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  Game Physics

  • David H. Eberly(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  600 Chapter 7 Linear Algebra 7.4.5 Dot Product, Cross Product, and Triple Products In the last section we defined the dot product of two vectors u and v and named it u · v. We saw its relationship to projection of one vector onto another and found it to be a useful concept for constructing an orthonormal set of vectors from a linearly independent set of vectors. Cross Product In three dimensions we have another type of product between vectors u and v that fits in naturally with the topics of the last section, namely, the cross product, a vector we will denote by u × v. This vector is required to be perpendicular to each of its two arguments, and its length is required to be the area of the parallelogram formed by its arguments. Figure 7.17 illustrates the cross product. Your intuition should tell you that there are infinitely many vectors that are per- pendicular to both u and v, all such vectors lying on a normal line to the plane spanned by u and v. Of these, only two have the required length. We need to make a selection between the two. The standard convention uses what is called the right-hand rule. If you place your right hand so the fingers point in the direction of u (the first argument) as shown Figure 7.17(a), then rotate your fingers towards v (the second argument) so that you make a closed fist, the cross product is selected to be that vector in which direction your thumb points. v u × v u (a) (b) v u |u| Figure 7.17 (a) The cross product of u and v according to the right-hand rule. (b) The paral- lelogram formed by u and v with angle θ and parallelogram base length and height marked. 7.4 Vector Spaces 601 The area of the parallelogram is α = bh where b is the length of the base (the length of u) and h is the height (the length of the projection of v onto a vector perpendicular to u): α = bh = |u||v| sin(θ) (7.17) Let w be a unit-length vector perpendicular to both u and v and in the direction con- sistent with the right-hand rule.

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  Linear Algebra

  • Elizabeth S. Meckes, Mark W. Meckes(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 Inner Products 4.1 Inner Products We saw in Section 1.3 that there were various ways in which the geometry of R n could shed light on linear systems of equations. We used a very limited amount of geometry, though; we only made use of general vector space operations. The geometry of R n is much richer than that of an arbitrary vector space because of the concepts of length and angles; it is extensions of these ideas that we will explore in this chapter. The Dot Product in R n Recall the following definition from Euclidean geometry. Definition Let x, y ∈ R n . The dot product or inner product of x and y is denoted x, y and is defined by x, y = n  j=1 x j y j , where x = ⎡ ⎢ ⎣ x 1 . . . x n ⎤ ⎥ ⎦ and y = ⎡ ⎢ ⎣ y 1 . . . y n ⎤ ⎥ ⎦ . Quick Exercise #1. Show that x, y = y T x for x, y ∈ R n . The dot product is intimately related to the ideas of length and angle: the length x of a vector x ∈ R n is given by x =  x 2 1 + · · · + x 2 n =  x, x, 226 Inner Products and the angle θ x,y between two vectors x and y is given by θ x,y = cos −1  x, y x y  . In particular, the dot product gives us a condition for perpendicularity: two vectors x, y ∈ R n are perpendicular if they meet at a right angle, which by the formula above is equivalent to the condition x, y = 0. For example, the standard basis vectors e 1 , . . . , e n ∈ R n are perpendicular to each other, since  e i , e j  = 0 for i  = j. Perpendicularity is an extremely useful concept in the context of linear algebra; the following proposition gives a first hint as to why. Proposition 4.1 Let (e 1 , . . . , e n ) denote the standard basis of R n . If v = ⎡ ⎢ ⎣ v 1 . . . v n ⎤ ⎥ ⎦ ∈ R n , then for each i, v i = v, e i  . Proof For v as above,  v, e j  =  k v k (e j ) k = v j , since e j has a 1 in the jth position and zeroes everywhere else.

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  640 CHAPTER 9 ■ Vectors in Two and Three Dimensions DEFINITION OF THE DOT PRODUCT If u  8 a 1 , a 2 9 and v  8 b 1 , b 2 9 are vectors, then their dot product, denoted by u # v, is defined by u # v  a 1 b 1  a 2 b 2 Thus to find the dot product of u and v, we multiply corresponding components and add. The dot product is not a vector; it is a real number, or scalar. EXAMPLE 1 ■ Calculating Dot Products (a) If u  3, 2 and v  4, 5 then u # v  1 3 21 4 2  1 2 21 5 2  2 (b) If u  2 i  j and v  5 i  6 j, then u # v  1 2 21 5 2  1 1 21 6 2  4 Now Try Exercises 5(a) and 11(a) ■ The proofs of the following properties of the dot product follow easily from the definition. PROPERTIES OF THE DOT PRODUCT 1. u # v  v # u 2. 1 c u 2 # v  c 1 u # v 2  u # 1 c v 2 3. 1 u  v 2 # w  u # w  v # w 4. 0 u 0 2  u # u Proof We prove only the last property. The proofs of the others are left as exercises. Let u  8 a 1 , a 2 9 . Then u # u  a 1 a 1  a 2 a 2  a 2 1  a 2 2  0 u 0 2 ■ Let u and v be vectors, and sketch them with initial points at the origin. We define the angle u between u and v to be the smaller of the angles formed by these represen- tations of u and v (see Figure 1). Thus 0  u  p. The next theorem relates the angle between two vectors to their dot product. THE DOT PRODUCT THEOREM If u is the angle between two nonzero vectors u and v, then u # v  0 u 00 v 0 cos u Proof Applying the Law of Cosines to triangle AOB in Figure 2 gives 0 u  v 0 2  0 u 0 2  0 v 0 2  2 0 u 00 v 0 cos u y x 0 v u ¨ FIGURE 1 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Elementary Linear Algebra, International Metric Edition

  • Ron Larson(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  5.1 Length and Dot Product in R n 5.2 Inner Product Spaces 5.3 Orthonormal Bases: Gram-Schmidt Process 5.4 Mathematical Models and Least Squares Analysis 5.5 Applications of Inner Product Spaces 231 5 Inner Product Spaces Electric/Magnetic Flux (p. 240) Heart Rhythm Analysis (p. 255) Revenue (p. 266) Torque (p. 277) Work (p. 248) Clockwise from top left, Jezper/Shutterstock.com; Lisa F. Young/Shutterstock.com; iStockphoto.com/kupicoo; Andrea Danti/Shutterstock.com; Sebastian Kaulitzki/Shutterstock.com Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 232 Chapter 5 Inner Product Spaces 5.1 Length and Dot Product in R n Find the length of a vector and find a unit vector. Find the distance between two vectors. Find a dot product and the angle between two vectors, determine orthogonality, and verify the Cauchy-Schwarz Inequality, the triangle inequality, and the Pythagorean Theorem. Use a matrix product to represent a dot product. VECTOR LENGTH AND UNIT VECTORS Section 4.1 mentioned that vectors can be characterized by two quantities, length and direction. This section defines these and other geometric properties (such as distance and angle) of vectors in R n . Section 5.2 extends these ideas to general vector spaces. You will begin by reviewing the definition of the length of a vector in R 2 . If v = ( v 1 , v 2 ) is a vector in R 2 , then the length , or norm , of v , denoted by H20648 v H20648 , is the length of the hypotenuse of a right triangle whose legs have lengths of uni2223 v 1 uni2223 and uni2223 v 2 uni2223 , as shown in Figure 5.1. Applying the Pythagorean Theorem produces H20648 v H20648 2 = uni2223 v 1 uni2223 2 + uni2223 v 2 uni2223 2 = v 1 2 + v 2 2 H20648 v H20648 = radical.alt2 v 1 2 + v 2 2 . Using R 2 as a model, the length of a vector in R n is defined below. This definition shows that the length of a vector cannot be negative.

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  Mathematical Methods in the Earth and Environmental Sciences

  • Adrian Burd(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  The converse of this is that three vectors are linearly independent if their scalar triple product is nonzero. Exercise 4.3.13 Show that if u is a linear combination of v and w, then u · (v × w) = 0. Exercise 4.3.14 Show that the vectors u = ˆ ı + ˆ j − ˆ k, v = 2ˆ ı − ˆ j + 3 ˆ k, and w = 3ˆ ı + 2 ˆ k are coplanar. The last vector product we will talk about is the vector triple product. This is something of a complicated beast. If we have three vectors A, B, and C, then the vector triple product is A × (B × C). The vector U = B × C is a vector that is perpendicular to both B and C. So, the vector A × (B × C) = A × U is a vector that is perpendicular to both A and B × C, and so must lie in the same plane as B and C (Figure 4.16). The vector triple product can be expressed as the difference of two vectors, A × (B × C) = (A · C)B − (A · B)C. (4.45) Equation (4.45) may look a little strange at first, but recall that the scalar product of two vectors is just a number, so the vector triple product A × (B × C) is a linear combination of the vectors B and C, which we can also see from Figure 4.16. As we can see from Equation (4.45), the placement of the parentheses is very important. In fact, because the vector cross 187 4.4 Matrices n A A p Figure 4.17 The projection of the vector A into a plane that has a unit normal n is A p . product does not commute (i.e., A × B = −B × A), we can see that, for example, A × (B × C) = −(B × C) × A. We can use the vector triple product to calculate the projection (A p ) of a vector A into a plane that has a unit normal vector n (Figure 4.17). The simplest way to calculate A p is to realize that the component of A perpendicular to the plane is just A · n, and then A p = A − (A · n)n. But we can also use the vector triple product, because using Equation (4.45) we find n × (A × n) = (n · n)A − (A · n)n = A − (A · n)n, which is the same formula we had before.

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