Vector Spaces In Physics - San Francisco State University
San Francisco State UniversityDepartment of Physics and AstronomyAugust 6, 2015Vector Spaces in PhysicsNotesforPh 385: Introduction toTheoretical Physics IR. Bland
TABLE OF CONTENTSChapter I. VectorsA. The displacement vector.B. Vector addition.C. Vector products.1. The scalar product.2. The vector product.D. Vectors in terms of components.E. Algebraic properties of vectors.1. Equality.2. Vector Addition.3. Multiplication of a vector by a scalar.4. The zero vector.5. The negative of a vector.6. Subtraction of vectors.7. Algebraic properties of vector addition.F. Properties of a vector space.G. Metric spaces and the scalar product.1. The scalar product.2. Definition of a metric space.H. The vector product.I. Dimensionality of a vector space and linear independence.J. Components in a rotated coordinate system.K. Other vector quantities.Chapter 2. The special symbols ij and ijk, the Einstein summationconvention, and some group theory.A. The Kronecker delta symbol, ijB. The Einstein summation convention.C. The Levi-Civita totally antisymmetric tensor.Groups.The permutation group.The Levi-Civita symbol.D. The cross Product.E. The triple scalar product.F. The triple vector product.The epsilon killer.Chapter 3. Linear equations and matrices.A. Linear independence of vectors.B. Definition of a matrix.C. The transpose of a matrix.D. The trace of a matrix.E. Addition of matrices and multiplication of a matrix by a scalar.F. Matrix multiplication.G. Properties of matrix multiplication.H. The unit matrixI. Square matrices as members of a group.ii
J. The determinant of a square matrix.K. The 3x3 determinant expressed as a triple scalar product.L. Other properties of determinantsProduct lawTranspose lawInterchanging columns or rowsEqual rows or columnsM. Cramer's rule for simultaneous linear equations.N. Condition for linear dependence.O. Eigenvectors and eigenvaluesChapter 4. Practical examplesA. Simple harmonic motion - a reviewB. Coupled oscillations - masses and springs.A system of two masses.Three interconnected masses.Systems of many coupled masses.C. The triple pendulumChapter 5. The inverse; numerical methodsA. The inverse of a square matrix.Definition of the inverse.Use of the inverse to solve matrix equations.The inverse matrix by the method of cofactors.B. Time required for numerical calculations.C. The Gauss-Jordan method for solving simultaneous linear equations.D. The Gauss-Jordan method for inverting a matrix.Chapter 6. Rotations and tensorsA. Rotation of axes.B. Some properties of rotation matrices.OrthogonalityDeterminantC. The rotation group.D. TensorsE. Coordinate transformation of an operator on a vector space.F. The conductivity tensor.G. The inertia tensor.Chapter 6a. Space-time four-vectors.A.B.C.D.E.F.G.The origins of special relativity.Four-vectors and invariant proper time.The Lorentz transformation.Space-time events.The time dilation.The Lorentz contraction.The Maxwell field tensor.iii
Chapter 7. The Wave EquationA. Qualitative properties of waves on a string.B. The wave equation.Partial derivatives.Wave velocity.C. Sinusoidal solutions.D. General traveling-wave solutions.E. Energy carried by waves on a string.Kinetic energy.Potential energy.F. The superposition principle.G. Group and phase velocity.Chapter 8. Standing Waves on a StringA. Boundary Conditions and Initial ConditionsString fixed at a boundary.Boundary between two different strings.B. Standing waves on a string.Chapter 9. Fourier SeriesA. The Fourier sine series.The general solution.Initial conditions.Orthogonality.Completeness.B. The Fourier sine-cosine Series.Odd and even functions.Periodic functions in time.C. The exponential Fourier series.Chapter 10. Fourier Transforms and the Dirac Delta FunctionA. The Fourier transform.B. The Dirac delta function (x).The rectangular delta function.The Gaussian delta function.Properties of the delta function.C. Application of the Dirac delta function to Fourier transforms.Basis states.Functions of position x.D. Relation to quantum mechanics.Chapter 11. Maxwell's Equations in Special RelativityAppendix A. Useful mathematical facts and formulae.A. Complex numbers.B. Some integrals and identitiesC. The small-angle approximation.D. Mathematical logical symbols.Appendix B. Using the P&A Computer System1. Logging on.2. Running MatLab, Mathematica and IDL.3. Mathematica.iv
Appendix C. Mathematica1. Calculation of the vector sum using Mathematica.2. Matrix operations in Mathematica3. Speed test for Mathematica.Referencesv
Vector Spaces in Physics8/6/2015Chapter 1. VectorsWe are all familiar with the distinction between things which have a direction and thosewhich don't. The velocity of the wind (see figure 1.1) is a classical example of a vectorFigure 1-1. Where is the vector?quantity. There are many more of interest in physics, and in this and subsequent chapterswe will try to exhibit the fundamental properties of vectors.Vectors are intimately related to the very nature of space. Euclidian geometry (planeand spherical geometry) was an early way of describing space. All the basic concepts ofEuclidian geometry can be expressed in terms of angles and distances. A more recentdevelopment in describing space was the introduction by Descartes of coordinates alongthree orthogonal axes. The modern use of Cartesian vectors provides the mathematicalbasis for much of physics.A. The Displacement VectorThe preceding discussion did not lead to a definition of a vector. But you can convinceyourself that all of the things we think of as vectors can be related to a single fundamentalquantity, the vector r representing the displacement from one point in space to another.Assuming we know how to measure distances and angles, we can define a displacementvector (in two dimensions) in terms of a distance (its magnitude), and an angle: displacement from r12 (1-1) point 1 to point 2 distance, angle measured counterclockwise from due East (See figure 1.2.) Note that to a given pair of points corresponds a unique displacement,but a given displacement can link many different pairs of points. Thus the fundamentaldefinition of a displacement gives just its magnitude and angle.We will use the definition above to discuss certain properties of vectors from a strictlygeometrical point of view. Later we will adopt the coordinate representation of vectorsfor a more general and somewhat more abstract discussion of vectors.1-1
Vector Spaces in Physics8/6/2015distance rpoint2angleeastpoint1Figure 1-2. A vector, specified by giving a distance and an angle.B. Vector AdditionA quantity related to the displacement vector is the position vector for a point.Positions are not absolute – they must be measured relative to a reference point. If wecall this point O (the "origin"), then the position vector for point P can be defined asfollows:rP displacement from point O to point P (1-2)It seems reasonable that the displacement from point 1 to point 2 should be expressed interms of the position vectors for 1 and 2. We are be tempted to write (1-3) r12 r2 r1A "difference law" like this is certainly valid for temperatures, or even for distances alonga road, if 1 and 2 are two points on the road. But what does subtraction mean forvectors? Do you subtract the lengths and angles, or what? When are two vectors equal?In order to answer these questions we need to systematically develop the algebraicproperties of vectors. We will let A , B , C , etc. represent vectors. For the moment, the only vectorquantities we have defined are displacements in space. Other vector quantities which wewill define later will obey the same rules.Definition of Vector Addition. The sum of two vector displacements can be defined soas to agree with our intuitive notions of displacements in space. We will define the sumof two displacements as the single displacement which has the same effect as carrying outthe two individual displacements, one after the other. To use this definition, we need tobe able to calculate the magnitude and angle of the sum vector. This is straightforwardusing the laws of plane geometry. (The laws of geometry become more complicated inthree dimensions, where the coordinate representation is more convenient.) Let A and B be two displacement vectors, each defined by giving its length and angle: A ( A, A ),(1-4) B ( B, B ).1-2
Vector Spaces in Physics8/6/2015Here we follow the convention of using the quantity A (without an arrow over it) to represent the magnitude of A ; and, as stated above, angles are measuredcounterclockwise from the easterly direction. Now imagine points 1, 2, and 3 such that A represents the displacement from 1 to 2, and B represents the displacement from 2 to3. This is illustrated in figure 1-3. B A3 B2 A1 Figure 1-3. Successive displacements A and B . Definition: The sum of two given displacements A and B is the third displacement C which has the same effect as making displacements A and B in succession. It is clear that the sum C exists, and we know how to find it. An example is shown in figure 1-4 with two given vectors A and B and their sum C . It is fairly clear that the length and angle of C can be determined (using trigonometry), since for the triangle 1-23, two sides and the included angle are known. The example below illustrates thiscalculation. Example: Let A and B be the two vectors shown in figure 1-4: A (10 m, 48 ), B (14 m, 20 ). Determine the magnitude and angle from due east of their sum C , where C A B . The angle opposite side C can be calculated asshown in figure 1-4; the result is that 2 152 . Then the length of side C can becalculated from the law of1-3
Vector Spaces in Physics8/6/2015 C A B A20 1 48 348 2 C 210 m3B14 m248 -20 28 320 180 -28 152 C11Figure 1-4. Example of vector addition. Each vector's direction is measured counterclockwise from due East. Vector A is a displacement of 10 m at an angle of 48 and vector B is a displacement of 14 m at anangle of 20 .cosines:C2 A2 B2 -2AB cos 2givingC [(10 m)2 (14 m)2 - 2(10m)(14m)cos 152 ]1/2 23.3072 m .The angle 1 can be calculated from the law of sines:sin 1 / B sin 2 / Cgiving 1 sin-1 .28200 16.380 .The angle C is then equal to 48 - 1 31.620 . The result is thusC 23.3072 m, 31.620 .One conclusion to be drawn from the previous example is that calculations using thegeometrical representation of vectors can be complicated and tedious. We will soon seethat the component representation for vectors simplifies things a great deal.C. Product of Two VectorsMultiplying two scalars together is a familiar and useful operation. Can we do thesame thing with vectors? Vectors are more complicated than scalars, but there are twouseful ways of defining a vector product.The Scalar Product. The scalar product, or dot product, combines two vectors to give ascalar: (1-5)A B A B cos( B - A )1-4
Vector Spaces in Physics8/6/2015This simple definition is illustrated in figure1-5. One special property of the dot product is its relation to the length A of a vector A : A A A2(1-6)This in fact can be taken as the definition ofthe length, or magnitude, A. B B - A AAn interesting and useful type of vector is a unit vector, defined as a vector of length 1.Figure 1-5. Vectors A and B . Their dotWe usually write the symbol for a unit vectorproduct is equal to A B cos( B - A). with a hat over it instead of a vector symbol:Their cross product A B has magnitudeû . Unit vectors always have the propertyA B sin( B - A), directed out of the paper.(1-7)uˆ uˆ 1Another use of the dot product is to define orthogonality of two vectors. If the anglebetween the two vectors is 90 , they are usually said to be perpendicular, or orthogonal.Since the cosine of 90 is equal to zero, we can equally well define orthogonality of twovectors using the dot product:(1-8)A B A B 0Example. One use of the scalar product in physics is in calculating the work done by a force F acting on a body while it is displaced by the vector displacement d . The work done depends on the distance moved, but in a specialway which projects out the distance moved in the direction of the force:(1-9)Work F d Similarly, the power produced by the motion of an applied force F whose pointof application is moving with velocity v is given by(1-10)Power F vIn both of these cases, two vectors are combined to produce a scalar.Example. To find the component of a vector A in a direction given by the unitvector n̂ , take the dot product of the two vectors.(1-11)Component of A along n A nˆThe Vector Product. The vector product, or cross product, is considerably morecomplicated than the scalar product. It involves the concept of left and right, which hasan interesting history in physics. Suppose you are trying to explain to someone on adistant planet which side of the road we drive on in the USA, so that they could build acar, bring it to San Francisco and drive around. Until the 1930's, it was thought that therewas no way to do this without referring to specific objects which we arbitrarily designateas left-handed or right-handed. Then it was shown, by Madame Wu, that both theelectron and the neutrino are intrinsically left-handed! This permits us to tell the alienhow to determine which is her right hand. "Put a sample of 60Co nuclei in front of you,on a mount where it can rotate freely about a vertical axis. Orient the nuclei in a1-5
Vector Spaces in Physics8/6/2015magnetic field until the majority of the decay electrons go downwards. The sample willgradually start to rotate so that the edge nearest you moves to the right. This is said to bea right-handed rotation about the vertically upward axis." The reason this works is thatthe magnetic field aligns the cobalt nuclei vertically, and the subsequent nuclear decaysemit electrons preferentially in the opposite direction to the nuclear spin. (Cobalt-60decays into nickel-60 plus an electron and an anti-electron neutrino,60(1-12)Co 60 Ni e νeSee the Feynman Lectures for more information on this subject.) Now you can just tellthe alien, "We drive on the right." (Hope she doesn't land in Australia.)z A B By Ax Figure 1-6. Illustration of a cross product. Can you prove that if A is along the x- axis, then A B is in the y-z plane? This lets us define the cross product of two vectors A and B as shown in figure 1-5. The cross product of these two vectors is a third vector C , with magnitudeC A B sin ( B - A) ,(1-13) perpendicular to the plane containing A and B , and in the sense "determined by theright-hand rule." This last phrase, in quotes, is shorthand for the following operational definition: Place A and B so that they both start at the same point. Choose a third direction perpendicular to both A and B (so far, there are two choices), and call it the upward direction. If, as A rotates towards B , it rotates in a right-handed direction, then this third direction is the direction of C .Example. The Lorentz Force is the force exerted on a charged particle due toelectric and magnetic fields. If the particle's charge is given by q and it is movingwith velocity v , in electric field E and magnetic field B , the force F is givenby(1-14)F qE qv BThe second term is an example of a vector quantity created from two othervectors.1-6
Vector Spaces in Physics8/6/2015Example. The cross product is used to find the direction of the third axis in athree-dimensional space. Let û and v̂ be two orthogonal unit vectors,representing the first (or x) axis and the second (or y) axis, respectively. A unitvector ŵ in the correct direction for the third (or z) axis of a right-handedcoordinate system is found using the cross product:ˆ uˆ vˆ(1-15)wD. Vectors in Terms of ComponentsUntil now we have discussed vectors from a purely geometrical point of view. There isanother representation, in terms of components, which makes both theoretical analysisand practical calculations easier. It is a fact about the space that we live in that it ispossible to find three, and no more than three, vectors that are mutually orthogonal. (Thisis the basis for calling our space three dimensional.) Descartes first introduced the ideaof measuring position in space by giving a distance along each of three such vectors. ACartesian coordinate system is determined by a particular choice of these three vectors.In addition to requiring the vectors to be mutually orthogonal, it is convenient to takeeach one to have unit length.A set of three unit vectors defining a Cartesian coordinate system can be chosen asfollows. Start with a unit vector iˆ in any direction you like. Then choose any secondunit vector ĵ which is perpendicular to iˆ . As the third unit vector, take kˆ iˆ ˆj .These three unit vectors (iˆ, ˆj, kˆ) are said to be orthonormal. This means that they aremutually orthogonal, and normalized so as to be unit vectors. We will often refer to theirdirections as the x, y, and z directions. We will also sometimes refer to the three vectorsas (eˆ1 , eˆ2 , eˆ3 ) , especially when we start writing sums over the three directions. Suppose that we have a vector A and three orthogonal unit vectors (iˆ, ˆj, kˆ) , all definedas in the previous sections by their length and direction. The three unit vectors can be used to define vector components of A , as follows: Ax A iˆ, Ay A ˆj ,(1-16) Az A kˆ.This suggests that we can start a discussion of vectors from a component view, by simplydefining vectors as triplets of scalar numbers: Ax Component Representation of Vectors(1-17)A Ay A z It remains to prove that this definition is completely equivalent to the geometricaldefinition, and to define vector addition and multiplication of a vector by a scalar in termsof components.1-7
Vector Spaces in Physics8/6/2015Let us show that these two ways of specifying a vector are equivalent - that is, to eachgeometrical vector (magnitude and direction) there corresponds a single set ofcomponents, and (conversely) to each set of components there corresponds a singlegeometrical vector. The first assertion follows from the relation (1-16), showing how todetermine the triplet of components for any given geometrical vector. The dot product ofany two vectors exists, and is unique. The converse is demonstrated in Figure 1-7. It is seen that the vector A can be written asthe sum of three vectors proportional to its three components: (1-18)A iˆA ˆjA kˆAxyzFrom the diagram it is clear that, given three components, there is just one such sum. So,we have established the equivalencezk̂AzyiˆAxĵAyx Figure 1-7. Illustration of the addition of the component vectors iAx, jAy, and kAz to get the vector A . This proves that a given set of values for (Ax, Ay, Az) leads to a unique vector A in the geometrical picture. Ax A (magnitude, direction) Ay A z (1-19)E. Algebraic Properties of Vectors.As a warm-up, consider the familiar algebraic properties of scalars. This provides a roadmap for defining the analogous properties for vectors.Equality.1-8
Vector Spaces in Physics8/6/2015a b b aa b and b c a cAddition and multiplication of scalars.a b b aa (b c) (a b) c a b cab baa(bc) (ab)c abca(b c) ab acZero, negative numbers.a 0 aa ( a) 0No surprises there.Equality. We will say that two vectors are equal, meaning that they are really the samevector, if all three of their components are equal:A B Ax Bx , Ay By , Az Bz .(1-20)The commutative property, A B B A , and the transitive property,A B and B C A C follow immediately, since components are scalars.Vector Addition. We will adopt the obvious definition of vector addition usingcomponents: C A B C x Ax B x DEFINITION (1-21) C y Ay B y C A B zz zThat is to say, the components of the sum are the sums of the components of the vectorsbeing added. It is necessary to show that this is in fact the same definition as the one weintroduced for geometrical vectors. This can be seen from the geometrical constructionshown in figure 1-7a.1-9
Vector Spaces in Physics8/6/2015zCByAxFigure 1.7a. The components of the sum vectorcomponents of the two vectors being summed.C are seen to be the algebraic sum of theMultiplication of a vector by a scalar. We will take the following, rather obvious, definition of multiplication of a vector A by a scalar c: cAx DEFINITION (1-23)cA cAy cA z It is pretty clear that this is consistent with the procedure in the geometricalrepresentation: just multiply the length by the scalar c, leaving the angle unchanged.The Zero VectorWe define the zero vector as follows: 0 DEFINITION (1-24)0 0 0 Taken with the definition of vector addition, it is clear that the essential relationA 0 A1 - 10
Vector Spaces in Physics8/6/2015is satisfied. And a vector with all zero-length components certainly fills the bill as thegeometrical version of the zero vector.The Negative of a Vector. The negative of a vector in terms of components is also easyto guess: Ax DEFINITION (1-26) A Ay A z The essential relation A A 0 will clearly be satisfied, in terms of components. It isalso easy to prove that this corresponds to the geometrical vector with the directionreversed; we will also omit this proof.Subtraction of Vectors. Subtraction is then defined byA B A ( B)subtraction of vectorsThat is, to subtract a vector from anotherone, just add the vector's negative. The"vector-subtraction parallelogram" forBtwo vectors A and B is shown in figure1-8. The challenge is to choose theAdirections of A and B such that thediagonal correctly represents head-to-tailBaddition of the vectors on the sides.(1-27) B AE. Algebraic properties of vectoraddition.A A BFigure 1-8. The vector-subtraction parallelogram.Can you put arrows on the sides of theparallelogram so that both triangles read as correctvector-addition equations?Vectors follow algebraic rules similar tothose for scalars:A B B Acommutative property of vector additionA ( B C ) ( A B) C ) a A B aA aB a b A aA aBc dA cd Aassociative property of vector additiondistributive property of scalar multiplication (1-28)another distributive propertyassociative property of scalar multiplicationIn the case of geometrically defined vectors, these properties are not so easy to prove,especially the second one. But for component vectors they all follow immediately (seethe problems). And so they must be correct also for geometrical vectors.As an illustration, we will prove the distributive law of scalar multiplication, above, forcomponent vectors. We use only properties of component vectors.1 - 11
Vector Spaces in Physics8/6/2015 Ax Bx a ( A B ) a Ay By definition of addition of vectors A B z z a Ax Bx a Ay By definition of multiplication by a scalar a Az Bz aAx aBx aAy aBy distributive property of scalar multiplication aA aB z z aAx aBx aAy aBy aA aB z z Ax Bx a Ay a By A B z z aA aBdefinition of addition of vectorsdefinition of multiplication of a vector by a scalarQEDThe proofs of the other four properties are similar.F. Properties of a Vector Space.Vectors are clearly important to physicists (and astronomers), but the simplicity andpower of representing quantities in terms of Cartesian components is such that vectorshave become a sort of mathematical paradigm. So, we will look in more detail at theirabstract properties, as members of a Vector Space.In Table 1-1 we give a summary of the basic properties which a set of objects must haveto constitute a vector space.1 - 12
Vector Spaces in Physics8/6/2015A vector space is a set of objects, called vectors, with the operations of addition oftwo vectors and multiplication of a vector by a scalar defined, satisfying thefollowing properties. 1. Closure under addition. If A and B are both vectors, then so is C A B. 2. Closure under scalar multiplication. If A is a vector and d is a scalar, thenB dA is a vector. 3. Existence of a zero. There exists a zero vector 0 , such that, for any vector A , A 0 A. 4. Existence of a negative. For any vector A there exists a negative A , such that A ( A) 0 .5. Algebraic Properties. Vector addition and scalar multiplication satisfy thefollowing rules:A B B A(1-29) commutativeA ( B C ) ( A B) C(1-30) associativea( A B) aA aB(1-31) distributive a b A aA bA(1-32) distributivec dA (cd ) A(1-33) associative Table 1-1. Properties of a vector space.Notice that in the preceding box, vectors are not specifically defined. Nor is the methodof adding them specified. We will see later that there are many different classes ofobjects which can be thought of as vectors, not just displacements or other threedimensional objects.Example: Check that the set of all component vectors, defined as triplets of realnumbers, does in fact satisfy all the requirements to constitute a vector space.Referring to Table 1-1, it is easy to see that the first four properties of a vectorspace are satisfied: Ax Bx 1. Closure under addition. If A Ay and B By are both vectors, then so A B z z Ax Bx is C A B Ay By . This follows from the fact that the sum of two scalars A B z zgives another scalar.1 - 13
Vector Spaces in Physics8/6/2015 2. Closure under multiplication. If A is a vector and d is a scalar, then dAx B dA dAy is a vector. This follows from the fact that the product of two dA z scalars gives another scalar. 0 3. Zero. There exists a zero vector 0 0 , such that, for any vector A , 0 Ax 0 x A 0 Ay 0 y A . This follows from the addition-of-zero property for A 0 z zscalars. Ax 4. Negative. For any vector A there exists a negative A Ay , such that A z A ( A) 0 . Adding components gives zero for the components of the sum.5. The algebraic properties (1-29) through (1-33) were discussed above;they are satisfied for component vectors.So, all the requirements for a vector space are satisfied by component vectors.This had better be true! The whole point of vector spaces is to generalize fromcomponent vectors in three-dimensional space to a broader category ofmathematical objects that are very useful in physics.Example: The properties above have clearly been chosen so that the usualdefinition of vectors, including how to add them, satisfies these conditions. Butthe concept of a vector space is intended to be more general. What if we definevectors in two dimensions geometrically (having a magnitude and an angle) andwe keep multiplication by a scalar the same, but we redefine vector addition in thefollowing way.C A B ( A B, A B )(1-34)This might look sort of reasonable, if you didn't know better. Which of theproperties (1)-(5) in Table 1-1 are satisfied?1. Closure under addition: OK. A B is an acceptable magnitude, and A B is an acceptable angle. (Angles greater than 360 are wrapped around.)2. Closure under scalar multiplication: OK3. Zero: the vector 0 (0,0) works fine; adding it on doesn't change A .4. Negative: Not OK. There is no way to add two positive magnitudes(magnitudes are non-negative) to get zero.5. Algebraic properties: You can easily show that these are all satisfied.1 - 14
Vector Spaces in Physics8/6/2015Conclusion: With this definition of vector addition, this is not a vector space.G. Metric Spaces and the Scalar ProductThe vector space as we have just defined it lacks something important. Thinking ofdisplacements, the length of a displacement, measuring the distance between two points,is essential to describing space. So we want to add a way of assigning a magnitude to avector. This is provided by the scalar product.The scalar product. The components of two vectors can be combined to give a scalar asfollows: A B Ax B x Ay B y Az B z DEFINITION(1-40) It is easy to show that the result follows from the representation of A and B in terms ofthe three unit vectors of the Cartesian coordinate system: A B (iˆAx ˆjAy kˆAz ) (iˆB x ˆjB y kˆBz ) Ax Bx Ay B y Az Bzwhere we have used the orthonormality of the unit vectors, iˆ iˆ ˆj ˆj kˆ kˆ 1 ,iˆ ˆj iˆ kˆ ˆj kˆ 0 . From the above definition in terms of components, it is easy todemonstrate the following algebraic properties for the scalar product:A B B A 1-13 A aB a A B A B C A B A C 1-14 1-15 The inner product of a displacement with itself can then be used to define a distancebetween points 1 and 2:r12 r12 r12 x122 y122 z122(1-41)From this expression we see that the distance between two points will not be zero unlessthey are in fact the same point.The scalar product can be used to define the direction cosines of an arbitrary vector,with respect to a set of Cartesian coordinate axes. The direction cosines are defined asfollows (see figure 1-9): DIRECTION COSINES , , and of a vector A(1-42)1 - 15
Vector Spaces in Physics8/6/20151 ˆ AxA i AA Ay1cos A ˆj AAA1 cos A kˆ zAASpecifying these three values is one way ofgiving the direction of a vector. However,only two angles are required to specify adirection in space, so these three angles mustnot be independent. It can be shown (seeproblems) thatcos2 cos2 cos2 1(1-43)cos z A y xFigure 1-9. The direction cosines for the vectorA are the cosines of the three angles shown.Definition of a Metric Space. The properties given in table 1-1 constitute the standarddefinition of a vector space. However, inclusion of a scalar product turns
5. The negative of a vector. 6. Subtraction of vectors. 7. Algebraic properties of vector addition. F. Properties of a vector space. G. Metric spaces and the scalar product. 1. The scalar product. 2. Definition of a metric space. H. The vector product. I. Dimensionality of a vector space and linear independence. J. Components in a rotated .
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The basic objects to be considered here are vector spaces of linear trans- formations, that is, a pair of vector spaces V and W and a linear subspace . The description of vector spaces of transformations of rank 1 is classical, . note will be concerned. Given any (abstract) vector space M, of dimension m, say, we may use the multiplication .
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12 VECTOR GEOMETRY 12.1 VectorsinthePlane Preliminary Questions 1. Answer true or false. Every nonzero vector is: (a) equivalent to a vector based at the origin. (b) equivalent to a unit vector based at the origin. (c) parallel to a vector based at the origin. (d) parallel to a unit vector based at the origin. solution (a) This statement is true. Translating the vector so that it is based on .
Vector Length (MVL) VEC-1 Typical MVL 64 (Cray) Add vector Typical MVL 64-128 Range 64-4996 (Vector-vector instruction shown) Vector processing exploits data parallelism by performing the same computation on linear arrays of numbers "vectors" using one instruction. The maximum number of elements in a vector supported by a vector ISA is
Have a brain storming session where they generate 10-15 themes. (Be patient, they'll get there eventually.) After they generate the list, allow people to talk in behalf of specific ideas Sometimes they may combine items Give everyone 3 votes and go through the list voting. If there is a clear winner then proceed. Otherwise repeat the process, but with one vote. Post the chosen theme .
