Scalar Triple Product

10 Key excerpts on "Scalar Triple Product"

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  Mathematical Methods in the Earth and Environmental Sciences

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

    (Publisher)

  The Scalar Triple Product has a nice cyclic symmetry, so that A · (B × C) = B · (C × A) = C · (A × B), and if we swap the order of the vector in any of the cross products, then we introduce a minus sign into the expression, so that A · (B × C) = −A(·C × B), etc. Exercise 4.3.11 Show that A · (B × C) = B · (C × A) = C · (A × B). Exercise 4.3.12 Show that A · (B × C) = −A(·C × B). The Scalar Triple Product gives us the volume of a parallelepiped (Figure 4.15) that is defined by the three vectors A, B, and C. Recall that the volume of a parallelepiped is the product of the area of the base and the perpendicular height of the parallelepiped. The vector product of B and C is a vector whose magnitude (  B × C ) is equal to the area of the base of the parallelepiped and that is orthogonal to both B and C. To get the perpendicular height of the volume, we take the scalar product of the vector A and the unit vector in the direction of B × C, that is the projection of A onto B × C: A · B × C  B × C . So, we end up with volume =  B × C  A · B × C  B × C  = A · (B × C). 186 Scalars, Vectors, and Matrices B C A B × C Figure 4.15 The Scalar Triple Product tells us the volume of a parallelepiped with vectors A, B, and C along its edges. B × C B C A A × (B × C) Figure 4.16 The vector triple product of three vectors A, B, and C. The vector B × C is perpendicular to the plane containing B and C, and the vector A × (B × C) is perpendicular to B × C and so lies in the plane containing B and C. One of the consequences of this equation is that if all the vectors A, B, and C lie in the same plane (i.e., they are coplanar), then the volume of the parallelepiped is zero (because the perpendicular height of the parallelepiped is zero), so A · (B × C) = 0, providing a nice test for coplanar vectors. The converse of this is that three vectors are linearly independent if their Scalar Triple Product is nonzero.

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  A Student's Guide to Vectors and Tensors

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

    (Publisher)

  Stated another way, if the projection of A onto the direction of B × C is not zero, then A cannot lie in the same plane as B and C . Thus A ◦ ( B × C ) = 0 (2.13) is both a necessary and a sufficient condition for vectors A , B , and C to be coplanar. Equating A ◦ ( B × C ) to the volume of the parallelepiped formed by vectors A , B , and C should also help you see that any cyclic permutation of the vectors (such as B ◦ ( C × A ) or C ◦ ( A × B ) ) gives the same result for the triple scalar product, since the volume of the parallelepiped is the same in each of these cases. Some authors describe this as the ability to interchange the dot and the cross without affecting the result (since ( A × B ) ◦ C is the same as C ◦ ( A × B ) ). One application in which the triple scalar product finds use is the determi-nation of reciprocal vectors, as explained in the sections in Chapter 4 dealing with covariant and contravariant components of vectors. 2.4 Triple vector product The triple scalar product described in the previous section is not the only use-ful way to multiply three vectors. An operation such as A × ( B × C ) (called the “triple vector product”) comes in very handy when you’re dealing with certain problems involving angular momentum and centripetal acceleration. Unlike the triple scalar product, which produces a scalar result (since the sec-ond operation is a dot product), the triple vector product yields a vector result 2.4 Triple vector product 33 (since both operations are cross products). You should note that A × ( B × C ) is not the same as ( A × B ) × C ; the location of the parentheses matters greatly in the triple vector product. The triple vector product is somewhat tedious to calculate by brute force, but thankfully a simplified expression exists: A × ( B × C ) = B ( A ◦ C ) − C ( A ◦ B ).

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  Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  It is not necessary to compute the dot product and cross product to evaluate a Scalar Triple Product—the value can be obtained directly from the formula u · (v × w) =       u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3       (9) the validity of which can be seen by writing u · (v × w) = u ·     v 2 v 3 w 2 w 3     i −     v 1 v 3 w 1 w 3     j +     v 1 v 2 w 1 w 2     k  = u 1     v 2 v 3 w 2 w 3     − u 2     v 1 v 3 w 1 w 3     + u 3     v 1 v 2 w 1 w 2     =       u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3       Example 5 Calculate the Scalar Triple Product u · (v × w) of the vectors u = 3i − 2 j − 5k, v = i + 4 j − 4k, w = 3 j + 2k Solution. u · (v × w) =       3 −2 −5 1 4 −4 0 3 2       = 49 TECHNOLOGY MASTERY Many calculating utilities have built-in cross product and determinant opera- tions. If your calculating utility has these capabilities, use it to check the computations in Examples 1 and 5. Figure 11.4.5 GEOMETRIC PROPERTIES OF THE Scalar Triple Product If u, v, and w are nonzero vectors in 3-space that are positioned so their initial points coincide, then these vectors form the adjacent sides of a parallelepiped (Figure 11.4.5). The following theorem establishes a relationship between the volume of this parallelepiped and the Scalar Triple Product of the sides. 706 Chapter 11 / Three-Dimensional Space; Vectors 11.4.6 THEOREM Let u, v, and w be nonzero vectors in 3-space. It follows from Formula (10) that u · (v × w) = ±V The + occurs when u makes an acute angle with v × w and the − occurs when it makes an obtuse angle. (a) The volume V of the parallelepiped that has u, v, and w as adjacent edges is V = |u · (v × w)| (10) (b) u · (v × w) = 0 if and only if u, v, and w lie in the same plane. PROOF (a) Referring to Figure 11.4.6, let us regard the base of the parallelepiped with Figure 11.4.6 u, v, and w as adjacent sides to be the parallelogram determined by v and w.

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  Understanding Geometric Algebra

  Hamilton, Grassmann, and Clifford for Computer Vision and Graphics

  • Kenichi Kanatani(Author)
  • 2015(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Today’s readers who are familiar with vector calculus would find his original formulation a little confusing, wondering why things that could be easily explained in terms of vector products and Scalar Triple Products are described in rather complicated forms. This is reasonable. As mentioned in the supplemental note to Chapter 2, today’s vector calculus was established by Gibbs, who combined and simplified Hamilton’s quaternion algebra in Chapter 4 and the Grassmann algebra in this chapter. In short, today’s vector calculus is a reformulation of the Grassmann algebra made easy in terms of vector products and Scalar Triple Products. Today, we specify the area and orientation of the parallelogram defined by vectors a and b by the vector product a × b and the volume and sign of the parallelepiped defined by vectors a , b , and c by the Scalar Triple Product | a , b , c | . As shown in Eq. (5.73), the dual ( a ∧ b ) ∗ is nothing but the vector product a × b , and the dual ( a ∧ b ∧ c ) ∗ is the Scalar Triple Product | a , b , c | . Today, as shown in Chapter 2, all quantities involving lines and 74 squaresolid Grassmann’s Outer Product Algebra planes in space are described by vectors and scalars, using vector products and Scalar Triple Products, which makes geometric descriptions and computations very easy. For this reason, the Grassmann algebra is seldom taught to students of physics and engineering today. However, vector calculus has a crucial restriction: it can be used only in 3D . In general n D, two vectors a and b span a 2D subspace. However, there exist infinitely many directions orthogonal to it, so we cannot define a unique surface normal to it. Hence, the only way to specify that plane is to write just a ∧ b . However, the same plane can be specified in many different ways, so various rules that the outer product ∧ should satisfy are obtained. Although this chapter only deals with 3D, all the descriptions can be extended to n D almost as is.

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  Vectors in Physics and Engineering

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

    (Publisher)

  2

  Vector algebra II Scalar and vector products

  Objectives After you have studied this chapter you should be able to

  • Use the geometric definition of the scalar product to calculate the scalar product of two given vectors (Objective 1 ).
  • Use the scalar product to determine the projection of a vector onto another vector (Objective 2 ).
  • Manipulate algebraic expressions involving the scalar product (Objective 3 ).
  • Test two given vectors for orthogonality (Objective 4 ).
  • Determine the scalar product of two cartesian vectors (Objective 5 ).
  • Use the scalar product to determine the magnitude of a given vector and the angle between two given vectors (Objective 6 ).
  • Use the geometric definition of the vector product to calculate the vector product of two given vectors (Objective 7 ).
  • Use the vector product to determine the areas of parallelograms and triangles (Objective 8 ).
  • Manipulate expressions containing vector products (Objective 9 ).
  • Determine the vector product of two cartesian vectors (Objective 10 ).
  • Use the vector product to test for collinear vectors (Objective 11 ).
  • Recognise and calculate Scalar Triple Products and vector triple products of given vectors (Objective 12 ).
  • Manipulate expressions involving triple products of vectors (Objective 13 ).
  • Use scalar products and vector products in a scientific and engineering context (Objective 14 ).

  Chapter 1 was concerned with multiplying a vector by a scalar (scaling) and the addition of vectors. This chapter is concerned with products of vectors. There are two useful ways in which a product can be formed from two vectors. One is called the scalar product because the product so formed is a scalar quantity; the other is called a 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

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  u ⋅ (v × w) = | | | | | | 3 −2 −5 1 4 −4 0 3 2 | | | | | | = 49 TECHNOLOGY MASTERY Many calculating utilities have built- in cross product and determinant op- erations. If your calculating utility has these capabilities, use it to check the computations in Examples 1 and 5. v w u Figure 11.4.5 GEOMETRIC PROPERTIES OF THE Scalar Triple Product If u, v, and w are nonzero vectors in 3-space that are positioned so their initial points co- incide, then these vectors form the adjacent sides of a parallelepiped (Figure 11.4.5). The following theorem establishes a relationship between the volume of this parallelepiped and the Scalar Triple Product of the sides. 706 Chapter 11 / Three-Dimensional Space; Vectors 11.4.6 theorem Let u, v, and w be nonzero vectors in 3-space. It follows from Formula (10) that u ⋅ (v × w) = ±V The + occurs when u makes an acute angle with v × w and the − occurs when it makes an obtuse angle. (a) The volume V of the parallelepiped that has u, v, and w as adjacent edges is V = |u ⋅ (v × w)| (10) (b) u ⋅ (v × w) = 0 if and only if u, v, and w lie in the same plane. proof (a) Referring to Figure 11.4.6, let us regard the base of the parallelepiped with v × w v w u h = || proj v×w u|| Figure 11.4.6 u, v, and w as adjacent sides to be the parallelogram determined by v and w. Thus, the area of the base is ‖v × w‖, and the altitude h of the parallelepiped (shown in the figure) is the length of the orthogonal projection of u on the vector v × w. Therefore, from Formula (12) of Section 11.3 we have h = ‖proj v×w u‖ = |u ⋅ (v × w)| ‖v × w‖ 2 ‖v × w‖ = |u ⋅ (v × w)| ‖v × w‖ It now follows that the volume of the parallelepiped is V = (area of base)(height) = ‖v × w‖h = |u ⋅ (v × w)| proof (b) The vectors u, v, and w lie in the same plane if and only if the parallelepiped with these vectors as adjacent sides has volume zero (why?). Thus, from part (a) the vectors lie in the same plane if and only if u ⋅ (v × w) = 0.

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  Calculus

  One and Several Variables

  • Saturnino L. Salas, Garret J. Etgen, Einar Hille(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  In this case, the first minus sign reverses the orientation of the triple and the second minus sign restores it. Finally, there are two distributive laws, the verification of which we postpone for a moment. (13.4.5) a × (b + c) = (a × b) + (a × c), (a + b) × c = (a × c) + (b × c). 666 ■ CHAPTER 13 VECTORS IN THREE-DIMENSIONAL SPACE The Scalar Triple Product Earlier we saw that (a, b, c) is a right-handed triple iff (a × b) · c > 0. The expression (a × b) · c is called a Scalar Triple Product. The absolute value of this number (it is a number, not a vector) has geometric significance. To describe it, we refer to Figure 13.4.6. There you see a parallelepiped with edges a, b, c. The absolute value of the Scalar Triple Product gives the volume of that parallelepiped: (13.4.6) V = |(a × b) · c|. a × b comp a × b c a b c Figure 13.4.6 PROOF The area of the base is a × b. (a × b was defined so that this would be the case.) The height of the parallelepiped is |comp a×b c|. Therefore V = |comp a×b c|a × b = |(a × b) · c| ❏ ↑ −−− (13.3.13) Of course, we could have formed the same parallelepiped using a different base (for example, using the vectors c and a) with a correspondingly different height (comp c×a b). Therefore, |(a × b) · c| = |(c × a) · b| = |(b × c) · a|. Since the a, b, c appear in the same cyclic order, the expressions inside the absolute value signs all have the same sign. (Property II of right-handed triples.) Therefore (13.4.7) (a × b) · c = (c × a) · b = (b × c) · a. Verification of the Distributive Laws We will verify the first distributive law, a × (b + c) = (a × b) + (a × c). The second follows readily from this one. The argument is left to you as an exercise. Take an arbitrary vector r and form the dot product [a × (b + c)] · r. We can then write [a × (b + c)] · r = (r × a) · (b + c) (13.4.7) = [(r × a) · b] + [(r × a) · c] = [(a × b) · r] + [(a × c) · r] (13.4.7) = [(a × b) + (a × c)] · r.

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  Fundamentals of University Mathematics

  • Colin McGregor, Jonathan Nimmo, Wilson Stothers(Authors)
  • 2010(Publication Date)
  • Woodhead Publishing

    (Publisher)

  V  = ±[a, b, c]. It has the positive sign if (a, b, c) is right-handed, and the negative sign otherwise. It remains to determine which triples give the plus sign and which the negative.

  We observe that, because the vectors axe in the same ‘cyclic order’, (v, w, u) and (w, u, v) determine parallelepipeds with the same orientation as the one determined by (u, v, w). On the other hand, (u, w, v), (v, u, w) and (w, v, u) determine parallelepipeds which are the ‘mirror image’ of the first set. It follows that

  as required.

  Remark As its proof suggests, Theorem 13.4.3 may be put more succinctly: a Scalar Triple Product is unchanged if we reorder the vectors so that they maintain their ‘cyclic order’ (see Figure 13.4.3 ), and otherwise the sign changes.

  Figure 13.4.3

  Theorem 13.4.4 If u = (

  ux

  ,

  uy

  ,

  uz

  ), v = (

  vx

  ,

  vy

  ,

  vz

  ) and w = (

  wx

  ,

  wy

  ,

  wz

  ) then

  (13.4.3)

  Proof This follows immediately from (13.2.8) ,

  Corollary 13.4.5 For a general 3 × 3 determinant Δ we have

  (1) Δ = 0 if any row is zero, (2) Δ = 0 if any row is a linear combination of the other rows, (3) If Δ′ is a determinant obtained by reordering the rows of Δ then Δ′ = Δ if the reordering is cyclic and Δ′ = – Δ otherwise.

  Proof This is simply a restatement of the properties of the Scalar Triple Product, making use of Theorem 13.4.4. We consider three arbitrary vectors u, v and w so that the determinant in (13.4.3) has arbitrary (real) entries.

  (1) From Definition 13.4.1 , [u, v, w] = 0 if any of u, v or w is zero.

  (2) If, for example, w = α u + v then u, v and w are coplanar and so, by Theorem 13.4.2 , [u, v, w] = 0.

  (3) This follows directly from Theorem 13.4.3 .

  Example 13.4.6

  (a) Show that (i, j, k ) is right-handed.

  (b) Let a = (1, 0, 1), b = (2, −1, 3) and c = (–2, 0, 0). Show that (a, b, c

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  The Mechanics and Thermodynamics of Continua

  • Morton E. Gurtin, Eliot Fried, Lallit Anand(Authors)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  The term point will be reserved for elements of E and the term vector for elements of the associated vector space V . Then: (i) The difference v = y − x between the points y and x is a vector. (ii) The sum y = x + v of a point x and a vector v is a point. (iii) Unlike the sum of two vectors, the sum of two points has no meaning. 1.1 Inner Product. Cross Product Our assumption that the point space E be Euclidean automatically endows the asso-ciated vector space V with an inner product. 3 We use the standard notation of vector analysis. In particular, • The inner product (a scalar) and cross product (a vector) 4 of vectors u and v are respectively designated by u · v and u × v . 3 The inner product is often referred to as the dot product. 4 We assume that the reader has some familiarity with these notions. The cross product is ordered in the sense that the cross product u × v of u and v is not generally equal to the cross product v × u of v and u . 3 4 Vector Algebra Figure 1.1. The parallelogram P defined by the vectors u and v and the direction of u × v determined by the right-hand screw rule. The inner product determines the magnitude (or length) of a vector u via the relation | u | = √ u · u ; and the angle θ = ∠ ( u , v ) (1.1) between nonzero vectors u and v is defined by cos θ = u · v | u || v | (0 ≤ θ ≤ π ) . (1.2) Since −| u || v | ≤ u · v ≤ | u || v | , this definition assigns exactly one angle θ to each pair of nonzero vectors u and v . Trivially, u · v = | u || v | cos θ ; this relation is often used to define the inner product. With regard to the cross product, the magnitude | u × v | (1.3) represents the area spanned by the vectors u and v ; that is, the area of the parallelogram P defined by these vectors as indicated in Figure 1.1 ; this area is nonzero if and only if u and v are linearly independent .

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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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