DOCSLIB.ORG

Concept of a Dyad and Dyadic: Consider Two Vectors a and B Dyad: It Consists of a Pair of Vectors a B for Two Vectors a a N D B

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Concept of a Dyad and Dyadic: Consider Two Vectors a and B Dyad: It Consists of a Pair of Vectors a B for Two Vectors a a N D B 1/11/2010 CHAPTER 1 Introductory Concepts • Elements of Vector Analysis • Newton’s Laws • Units • The basis of Newtonian Mechanics • D’Alembert’s Principle 1 Science of Mechanics: It is concerned with the motion of material bodies. • Bodies have different scales: Microscropic, macroscopic and astronomic scales. In mechanics - mostly macroscopic bodies are considered. • Speed of motion - serves as another important variable - small and high (approaching speed of light). 2 1 1/11/2010 • In Newtonian mechanics - study motion of bodies much bigger than particles at atomic scale, and moving at relative motions (speeds) much smaller than the speed of light. • Two general approaches: – Vectorial dynamics: uses Newton’s laws to write the equations of motion of a system, motion is described in physical coordinates and their derivatives; – Analytical dynamics: uses energy like quantities to define the equations of motion, uses the generalized coordinates to describe motion. 3 1.1 Vector Analysis: • Scalars, vectors, tensors: – Scalar: It is a quantity expressible by a single real number. Examples include: mass, time, temperature, energy, etc. – Vector: It is a quantity which needs both direction and magnitude for complete specification. – Actually (mathematically), it must also have certain transformation properties. 4 2 1/11/2010 These properties are: vector magnitude remains unchanged under rotation of axes. ex: force, moment of a force, velocity, acceleration, etc. – geometrically, vectors are shown or depicted as directed line segments of proper magnitude and direction. 5 e (unit vector) A A = A e – if we use a coordinate system, we define a basis set (iˆ , ˆj , k ˆ ): we can write A = Axi + Ay j + Azk Z or, we can also use the A three components and Y define X T {A}={Ax,Ay,Az} 6 3 1/11/2010 – The three components Ax , Ay , Az can be used as 3-dimensional vector elements to specify the vector. – Then, laws of vector-matrix algebra apply. – Tensors: scalar - an array of zero dimension vector - an array of one dimension 7 – quantities which need arrays of two or higher dimension to specify them completely - called tensors of appropriate rank. Again - to be a tensor, the object must also satisfy certain transformation properties of rotation and translation. Exs: Second-order tensors: stress at a point in deformable body - stress tensor has nine components (a 3x3 matrix in a representation when the basis is defined), inertia tensor (again, a 3x3 matrix in usual notation) expressing mass distribution in a rigid body.8 4 1/11/2010 • TYPES OF VECTORS: Consider a force F acting on a body at point B P. The force has a line F of action AB. This force P can lead to translation of the rigid body, rotation A of the rigid body about some point, as well as deformation of the body. 9 F F or P2 P1 The same force F is now acting at two different points P1 , P2 of the body, i.e., the lines of action are distinct. – same translational effect – the translational effect depends only on magnitude and direction of the force, not on its point of application or the line of action- free vectors 10 5 1/11/2010 B B F P1 F or P A A 2 The force F has the same line of action AB in the two cases. The points of application (P1 and P2) are different but moment about every point is the same → same rotational effect (as well as translational effect): effect of vector F depends on magnitude, direction as well as line of action - sliding vectors 11 If the body is deformable, the effect of force is different depending its point of application; whether the force acts at point P1 or P2. Thus, in such a case, the point of application is also crucial - bound vectors. B B F P or 1 F P A A 2 deformation 12 6 1/11/2010 • Equality of vectors: For free vectors A a n d B , A = B if and only if A a n d B have the same magnitude and the same direction. • Unit vectors: If A is a vector with magnitude A, A / A is a vector along A with unit length → e A = A /A or A =AeA. 13 e unit vector in A |1| eA the direction A = A e of A. • Addition of vectors: Consider two vectors A and B. Their addition is a vector C given by C = A + B . Also C = B + A (addition is commutative). The result is also a vector. 14 7 1/11/2010 C = A+B Graphically, one can use the parallelogram rule of vector addition. B C B For more than two vectors, one can add A sequentially - polygon of vectors. Consider The addition of vectors A, B, and D . C = A + B , E = (A + B) + D = C + D D or B E E = (A+ D)+ B C B = A+(B+ D). O A 15 • COMPONENTS OF A VECTOR: Consider the vector addition for A and B: graphically: C = A + B C B O A We can interpret A and B to be components of the vector C . Clearly, the components of C are non-unique. As an F B example, E and F are E C also components. O A We can make it more systematic. 16 8 1/11/2010 • Let e 1 , e 2 , e 3 - three linearly independent unit vectors (not necessarily orthogonal) and let A be a vector. We can write A = A 1 + A 2 + A 3 where Ai , i = 1, 2, 3 are components of A along the directions specified by unit vectors e i ,i = 1, 2, 3. e3 A Then : e 2 A3 A2 A= A1e1 +A2e2 +A3e3 A1 e1 A , i = 1, 2, 3 are unique scalar components i 17 Let B = B 1 e 1 + B 2 e 2 + B 3 e 3 be another vector, with components expressed in same unit vectors. We can then write the sum as C= A+B=(A1 +B1)e1 +(A2 +B2)e2 + (A3 + B3)e3 The components of the vector C are then → C1 = A1 + B1 , C2 = A2 + B2 and C3 = A3 + B3. 18 9 1/11/2010 • The more familiar case of unit vectors is the Cartesian coordinate system - (x, y, z) Let i , j , k - unit vectors along x, y and z directions. Then A=Ax i+Ay j+A2 k where Ax , Ay , Az are components of A Z A k along axes. z A Ay j Y Ax i X 19 •SCALAR PRODUCT: Definition: (DOT) For two vectors A and B, the dot product is defines as A ⋅ B = A B c o s θ Dot product is Commutative, i.e., A ⋅B = B⋅ A If 3 3 B A = Ai ei , B = Bi ei , i=1 i=1 θ A then A⋅B = A1B1 + A2B2 + A3B3 |B|cosθ provided the unit vectors are an orthogonal set, i.e., e1 ⋅ e 2 = e 2 ⋅ e3 = e3 ⋅ e1 = 0. 20 10 1/11/2010 •VECTOR PRODUCT: Let A , B be two vectors that make an angle θ with respect to each other. Then, the vector or cross product is defined as a vector C with magnitude C = A× B = A B sinθ . B Let k be the unit vector normal to the k θ plane formed by vectors A and B. It is A fixed by the right hand screw rule. Then A× B = A B sinθk 21 Some properties of cross product are : A×B=-B×A Consider unit vectors for the Cartesian coordinate system (x,y,z), (i, j, k): Then i i × j = j × k = k × i = 1 i × i = j × j = k × k = 0 k j i × j = k ; j× k = i right-hand rule k × i = j 22 11 1/11/2010 Now, consider cross-product again. When vectors A and B are expressed in component form: A=Ax i +Ay j+Az k B= Bxi+By j+Bzk, The cross product is evaluated by the operation i j k A × B = A x A y A z B x B y B z ≡ (Ay Bz − Az By )i + (Az Bx − Ax Bz ) j + (Ax By − Ay Bx )k 23 • SCALAR TRIPLE PRODUCT: Consider three vectors A, B, and C. The scalar triple product is given by R=A •(B×C) = (A×B)• C Note that the result is the same scalar quantity. It can be interpreted as the A volume of the C parallelepiped having the vectors A , B a n d C as B the edges. The sign can be +ve or -ve. 24 12 1/11/2010 • VECTOR TRIPLE PRODUCT: Consider vectors A, B, and C. Then, vector triple product is defined as a vector D, given by D = A × ( B × C ) Note that A×(B×C) ≠ (A×B)×C One can show that D=A×(B×C)=(A•C)B-(A•B)C • DERIVATIVE OF A VECTOR: Suppose that a vector A is a function of a scalar u, i.e., A = A ( u ) . We can then consider change in vector A associated with change in the scalar u. 25 Let A = A ( u ) and A (u + ∆u) ≡ A(u) + ∆A dA A(u) + ∆A -A(u) Then = lim du ∆u→0 ∆u dA ∆A or = lim du ∆u→0 ∆u This is the derivative of A with respect to u. Ex: The position vector r(t) for a particle moving depends on time. We define the dr ∆r velocity to be v(t) ≡ = lim dt ∆t→0 ∆t 26 13 1/11/2010 s ex: O’ z ∆r O P P’ r (s) x y OP ∆s Consider a particle P moving along a curved path.
Load more

Recommended publications
  • Relativistic Dynamics
    Chapter 4 Relativistic dynamics We have seen in the previous lectures that our relativity postulates suggest that the most efficient (lazy but smart) approach to relativistic physics is in terms of 4-vectors, and that velocities never exceed c in magnitude. In this chapter we will see how this 4-vector approach works for dynamics, i.e., for the interplay between motion and forces. A particle subject to forces will undergo non-inertial motion. According to Newton, there is a simple (3-vector) relation between force and acceleration, f~ = m~a; (4.0.1) where acceleration is the second time derivative of position, d~v d2~x ~a = = : (4.0.2) dt dt2 There is just one problem with these relations | they are wrong! Newtonian dynamics is a good approximation when velocities are very small compared to c, but outside of this regime the relation (4.0.1) is simply incorrect. In particular, these relations are inconsistent with our relativity postu- lates. To see this, it is sufficient to note that Newton's equations (4.0.1) and (4.0.2) predict that a particle subject to a constant force (and initially at rest) will acquire a velocity which can become arbitrarily large, Z t ~ d~v 0 f ~v(t) = 0 dt = t ! 1 as t ! 1 . (4.0.3) 0 dt m This flatly contradicts the prediction of special relativity (and causality) that no signal can propagate faster than c. Our task is to understand how to formulate the dynamics of non-inertial particles in a manner which is consistent with our relativity postulates (and then verify that it matches observation, including in the non-relativistic regime).
    [Show full text]
  • 21. Orthonormal Bases
    21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else.
    [Show full text]
  • S-96.510 Advanced Field Theory Course for Graduate Students Lecture Viewgraphs, Fall Term 2004
    S-96.510 Advanced Field Theory Course for graduate students Lecture viewgraphs, fall term 2004 I.V.Lindell Helsinki University of Technology Electromagnetics Laboratory Espoo, Finland I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.01 1 Contents [01] Complex Vectors and Dyadics [02] Dyadic Algebra [03] Basic Electromagnetic Equations [04] Conditions for Fields and Media [05] Duality Transformation [06] Affine Transformation [07] Electromagnetic Field Solutions [08] Singularities and Complex Sources [09] Plane Waves [10] Source Equivalence [11] Huygens’ Principle [12] Field Decompositions Vector Formulas, Dyadic Identites as an appendix I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.02 2 Foreword This lecture material contains all viewgraphs associated with the gradu- ate course S-96.510 Advanced Field Theory given at the Department of Electrical and Communications Engineering, fall 2004. The course is based on Chapters 1–6 of the book Methods for Electromagnetic Field Analysis (Oxford University Press 1992, 2nd edition IEEE Press, 1995, 3rd printing Wiley 2002) by this author. The figures drawn by hand on the blackboard could not be added to the present material. Otaniemi, September 13 2004 I.V. Lindell I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.03 3 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.04 4 S-96.510 Advanced Field Theory 1. Complex Vectors and Dyadics I.V.Lindell I.V.Lindell: Advanced Field Theory, 2004
    [Show full text]
  • Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
    Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi.
    [Show full text]
  • More on Vectors Math 122 Calculus III D Joyce, Fall 2012
    More on Vectors Math 122 Calculus III D Joyce, Fall 2012 Unit vectors. A unit vector is a vector whose length is 1. If a unit vector u in the plane R2 is placed in standard position with its tail at the origin, then it's head will land on the unit circle x2 + y2 = 1. Every point on the unit circle (x; y) is of the form (cos θ; sin θ) where θ is the angle measured from the positive x-axis in the counterclockwise direction. u=(x;y)=(cos θ; sin θ) 7 '$θ q &% Thus, every unit vector in the plane is of the form u = (cos θ; sin θ). We can interpret unit vectors as being directions, and we can use them in place of angles since they carry the same information as an angle. In three dimensions, we also use unit vectors and they will still signify directions. Unit 3 vectors in R correspond to points on the sphere because if u = (u1; u2; u3) is a unit vector, 2 2 2 3 then u1 + u2 + u3 = 1. Each unit vector in R carries more information than just one angle since, if you want to name a point on a sphere, you need to give two angles, longitude and latitude. Now that we have unit vectors, we can treat every vector v as a length and a direction. The length of v is kvk, of course. And its direction is the unit vector u in the same direction which can be found by v u = : kvk The vector v can be reconstituted from its length and direction by multiplying v = kvk u.
    [Show full text]
  • Glossary of Linear Algebra Terms
    INNER PRODUCT SPACES AND THE GRAM-SCHMIDT PROCESS A. HAVENS 1. The Dot Product and Orthogonality 1.1. Review of the Dot Product. We first recall the notion of the dot product, which gives us a familiar example of an inner product structure on the real vector spaces Rn. This product is connected to the Euclidean geometry of Rn, via lengths and angles measured in Rn. Later, we will introduce inner product spaces in general, and use their structure to define general notions of length and angle on other vector spaces. Definition 1.1. The dot product of real n-vectors in the Euclidean vector space Rn is the scalar product · : Rn × Rn ! R given by the rule n n ! n X X X (u; v) = uiei; viei 7! uivi : i=1 i=1 i n Here BS := (e1;:::; en) is the standard basis of R . With respect to our conventions on basis and matrix multiplication, we may also express the dot product as the matrix-vector product 2 3 v1 6 7 t î ó 6 . 7 u v = u1 : : : un 6 . 7 : 4 5 vn It is a good exercise to verify the following proposition. Proposition 1.1. Let u; v; w 2 Rn be any real n-vectors, and s; t 2 R be any scalars. The Euclidean dot product (u; v) 7! u · v satisfies the following properties. (i:) The dot product is symmetric: u · v = v · u. (ii:) The dot product is bilinear: • (su) · v = s(u · v) = u · (sv), • (u + v) · w = u · w + v · w.
    [Show full text]
  • 28. Exterior Powers
    28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1.
    [Show full text]
  • Lecture 3.Pdf
    ENGR-1100 Introduction to Engineering Analysis Lecture 3 POSITION VECTORS & FORCE VECTORS Today’s Objectives: Students will be able to : a) Represent a position vector in Cartesian coordinate form, from given geometry. In-Class Activities: • Applications / b) Represent a force vector directed along Relevance a line. • Write Position Vectors • Write a Force Vector along a line 1 DOT PRODUCT Today’s Objective: Students will be able to use the vector dot product to: a) determine an angle between In-Class Activities: two vectors, and, •Applications / Relevance b) determine the projection of a vector • Dot product - Definition along a specified line. • Angle Determination • Determining the Projection APPLICATIONS This ship’s mooring line, connected to the bow, can be represented as a Cartesian vector. What are the forces in the mooring line and how do we find their directions? Why would we want to know these things? 2 APPLICATIONS (continued) This awning is held up by three chains. What are the forces in the chains and how do we find their directions? Why would we want to know these things? POSITION VECTOR A position vector is defined as a fixed vector that locates a point in space relative to another point. Consider two points, A and B, in 3-D space. Let their coordinates be (XA, YA, ZA) and (XB, YB, ZB), respectively. 3 POSITION VECTOR The position vector directed from A to B, rAB , is defined as rAB = {( XB –XA ) i + ( YB –YA ) j + ( ZB –ZA ) k }m Please note that B is the ending point and A is the starting point.
    [Show full text]
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • The Dot Product
    The Dot Product In this section, we will now concentrate on the vector operation called the dot product. The dot product of two vectors will produce a scalar instead of a vector as in the other operations that we examined in the previous section. The dot product is equal to the sum of the product of the horizontal components and the product of the vertical components. If v = a1 i + b1 j and w = a2 i + b2 j are vectors then their dot product is given by: v · w = a1 a2 + b1 b2 Properties of the Dot Product If u, v, and w are vectors and c is a scalar then: u · v = v · u u · (v + w) = u · v + u · w 0 · v = 0 v · v = || v || 2 (cu) · v = c(u · v) = u · (cv) Example 1: If v = 5i + 2j and w = 3i – 7j then find v · w. Solution: v · w = a1 a2 + b1 b2 v · w = (5)(3) + (2)(-7) v · w = 15 – 14 v · w = 1 Example 2: If u = –i + 3j, v = 7i – 4j and w = 2i + j then find (3u) · (v + w). Solution: Find 3u 3u = 3(–i + 3j) 3u = –3i + 9j Find v + w v + w = (7i – 4j) + (2i + j) v + w = (7 + 2) i + (–4 + 1) j v + w = 9i – 3j Example 2 (Continued): Find the dot product between (3u) and (v + w) (3u) · (v + w) = (–3i + 9j) · (9i – 3j) (3u) · (v + w) = (–3)(9) + (9)(-3) (3u) · (v + w) = –27 – 27 (3u) · (v + w) = –54 An alternate formula for the dot product is available by using the angle between the two vectors.
    [Show full text]
  • Math 395. Tensor Products and Bases Let V and V Be Finite-Dimensional
    Math 395. Tensor products and bases Let V and V 0 be finite-dimensional vector spaces over a field F . Recall that a tensor product of V and V 0 is a pait (T, t) consisting of a vector space T over F and a bilinear pairing t : V × V 0 → T with the following universal property: for any bilinear pairing B : V × V 0 → W to any vector space W over F , there exists a unique linear map L : T → W such that B = L ◦ t. Roughly speaking, t “uniquely linearizes” all bilinear pairings of V and V 0 into arbitrary F -vector spaces. In class it was proved that if (T, t) and (T 0, t0) are two tensor products of V and V 0, then there exists a unique linear isomorphism T ' T 0 carrying t and t0 (and vice-versa). In this sense, the tensor product of V and V 0 (equipped with its “universal” bilinear pairing from V × V 0!) is unique up to unique isomorphism, and so we may speak of “the” tensor product of V and V 0. You must never forget to think about the data of t when you contemplate the tensor product of V and V 0: it is the pair (T, t) and not merely the underlying vector space T that is the focus of interest. In this handout, we review a method of construction of tensor products (there is another method that involved no choices, but is horribly “big”-looking and is needed when considering modules over commutative rings) and we work out some examples related to the construction.
    [Show full text]
  • Derivation of the Two-Dimensional Dot Product
    Derivation of the two-dimensional dot product Content 1. Motivation ....................................................................................................................................... 1 2. Derivation of the dot product in R2 ................................................................................................. 1 2.1. Area of a rectangle ...................................................................................................................... 2 2.2. Area of a right-angled triangle .................................................................................................... 2 2.3. Pythagorean Theorem ................................................................................................................. 2 2.4. Area of a general triangle using the law of cosines ..................................................................... 3 2.5. Derivation of the dot product from the law of cosines ............................................................... 5 3. Geometric Interpretation ................................................................................................................ 5 3.1. Basics ........................................................................................................................................... 5 3.2. Projection .................................................................................................................................... 6 4. Summary.........................................................................................................................................
    [Show full text]