Classical Mechanics Gregory R.D

Mathematics knows a several ways the multiplication of two, three or more vectors , all explained here. The two multiplying of the first group are the scalar and vector product – named according to the results, or the dot and a cross product of the vector – called according to the operation signs. We will consider them mainly geometrically, together with the 'communicator multiplication' – my private from the same group. Other multiplications have more vectors and are defined by the preceding ones, aggregating the factors as additional vectors. The text is an easier excerpt of the book of Quantum Mechanics, which this article is a promotion. 1 Scalar product The scalar product of two vectors, two oriented segments, a and b is a scalar a ⋅ b = ab cos γ, (1) where a = a and b = b are the intensities (lengths) of these vectors, and γ = ∠(a, b) is the angle between them. In other words, the scalar product is the product of the length of the first vector and the length of the projection of the second vector at first, as seen in the figure 1. In the wider theory of vectors, the scalar product is often called internal product, and dot product too.

2004

Commutative Law for Addition 2. A+ (B+C) _ (A+B) + C Associative Law for Addition 3. mA = Am Commutative Law for Multiplication 4. m (nA) _ (mn) A Associative Law for Multiplication 5. (m+ n) A = mA + nA Distributive Law 6. m (A+ B) = mA + mB Distributive Law Note that in these laws only multiplication of a vector by one or more scalars is used. In Chapter 2, products of vectors are defined. These laws enable us to treat vector equations in the same way as ordinary algebraic equations. For example, if A+B = C then by transposing A = C-B. A UNIT VECTOR is a vector having unit magnitude, if A is a vector with magnitude A 0, then A/A is a unit vector having the same-direction as A. Any vector A can be represented by a unit vector a in the direction of A multiplied by the magnitude of A. In symbols, A = Aa. THE RECTANGULAR UNIT VECTORS i, j, k. An important set of unit vectors are those having the directions of the positive x, y, and z axes of a three dimensional rectangular coordinate system, and are denoted respectively by i, j, and k (Fig.5). We shall use right-handed rectangular coordinate systems unless otherwise stated. Such a system derives z Fig. 5 Y VECTORS and SCALARS its name from the fact that a right threaded screw rotated through 900 from Ox to Oy will advance in the positive z direction, as in Fig.5 above. In general, three vectors A, B and C which have coincident initial points and are not coplanar, i.e. do not lie in or are not parallel to the same plane, are said to form a right-handed system or dextral system if a right threaded screw rotated through an angle less than 180° from A to B will advance in the direction C as shown in Fig.6. COMPONENTS OF A VECTOR. Any vector A in 3 dimensions can a represented with initial point at the origin 0 of a rec angular coordinate system (Fig.7). Let (Al, A2, A3) be the rectangular coordinates of the terminal point of vector A with initial point at 0. The vectors Ali, A2j, and A3k are called the recta lar component vectors or simply component vectors of A in the x, y and z directions respectively. A1, A2 and A3 are called the rectangular components or simply components of A in the x, y and z directions respectively. 43. Simplify 2A + B + 3C-{ A-2B-2 (2A-3B-C) }. Ans. 5A-3B + C 44. If a and b are non-collinear vectors and A = (x + 4y) a + (2x+y+1)b and B = (y-2x+2)a+ (2x-3y-1)b, find x and y such that 3A = 2B. Ans. x=2, y=-1 1001b 45. The base vectors a1, a2, a3 are given in terms of the base vectors b1, b2, b3 by the relations a1 = 2b1 + 3b2-b3 , a2 = b1-2b2 + 2b3 , a3 =-2b1 + b2-2b3 If F = 3b1-b2 + 2b3 , express F in terms of a1, a2 and a3. Ans. 2a1 + 5a2 + 3a3 46. If a, b, c are non-coplanar vectors determine whether the vectors r1 = 2a-3b + c , r2 = 3a-5b + 2c , and r3 = 4a-5b+ c are linearly independent or dependent. Ans. Linearly dependent since r3 = 5r1-2r2. 47. If A and B are given vectors representing the diagonals of a parallelogram, construct the parallelogram. 48. Prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and has one half of its magnitude. 49. (a) If 0 is any point within triangle ABC and P, Q, R are midpoints of the sides AB, BC, CA respectively, prove that OA + OB + OC = OP + OQ + OR. (b) Does the result hold if 0 is any point outside the triangle? Prove your result. Ans. Yes 50. In the adjoining figure, ABCD is a parallelogram with P and Q the midpoints of sides BC and CD respectively. Prove that AP and AQ trisect diagonal BD at the points E and F. 51. Prove that the medians of a triangle meet in a common point which is a point of trisection of the medians. 52. Prove that the angle bisectors of a triangle meet in a common point. 53. Show that there exists a triangle with sides which are equal and parallel to the medians of any given triangle. 54. Let the position vectors of points P and Q relative to an origin 0 be given by p and q respectively. If R is a point which divides line PQ into segments which are in the ratio m : n show that the position vector of R VECTORS and SCALARS 15 is given by r = 'nP +nq and that this is independent of the origin. m+n 55. If r1, r2, ..., rn are the position vectors of masses m1 , m2, ..., mn respectively relative to an origin 0, show that the position vector of the centroid is given by r = and that this is independent of the origin. m1r1+m2r2+...+mnrn In 1+m2+...+Inn 56. A quadrilateral ABCD has masses of 1, 2, 3 and 4 units located respectively at its vertices A (-1,-2, 2), B(3, 2,-1), C(1,-2, 4), and D(3, 1, 2). Find the coordinates of the centroid. Ans. (2, 0, 2) 57. Show that the equation of a plane which passes through three given points A, B, C not in the same straight line and having position vectors a, b, c relative to an origin 0, can be written r ma + nb + pc = m+ n+ p where in, n, p are scalars. Verify that the equation is independent of the origin. 58. The position vectors of points P and Q are given by r1 = 2i + 3j-k, r2 = 4i-3j + 2k. Determine PQ in terms of i, j, k and find its magnitude. Ans. 2i-6j + 3k, 7 59. If A= 3i-j-4k, B =-2i + 4j-3k, C =i + 2j-k, find (a) 2A-B +3C, (b) f A +B +C I, (c) 13A-2B +4C 1, (d) a unit vector parallel to 3A-2B +4C. (a) 11i-8k (b) (c) (d) 3A-2B + 4C Ans. 60. The following forces act on a particle P : F1 = 2i + 3j-5k, F2 =-5i + j + 3k, F3 = i-2j + 4k, F4 = 4i-3j-2k, measured in pounds. Find (a) the resultant of the forces, (b) the magnitude of the resultant. Ans. (a) 2i-j (b) yr 61. In each case determine whether the vectors are linearly independent or linearly dependent: (a) A=21+j-3k, B=i-4k, C=4i+3j-k, (b) A=i-3j+2k, B=2i-4j-k, C=3i+2j-k. Ans. (a) linearly dependent, (b) linearly independent 62. Prove that any four vectors in three dimensions must be linearly dependent. 63. Show that a necessary and sufficient condition that the vectors A = A 1 i + A 2 j + A3 k, B = B1 i + B2 j + B3 k, Al A2 A3 C=C I i +C2j +C3k be linearly independent is that the determinant B1 B2 B. be different from zero. C1 C2 C3 64. (a) Prove that the vectors A = 3i + j-2k, B =-i + 3j + 4k, C = 4i-2j-6k can form the sides of a triangle. (b) Find the lengths of the medians of the triangle. Ans. (b) vim, 2 v 4, 2 V-1-50 65. Given the scalar field defined by c(x, y, z) = 4yz3 + 3xyz-z2 + 2. Find (a) 0(1,-1,-2), (b) 4(0,-3,1). Ans. (a) 36 (b)-11 66. Graph the vector fields defined by (a) V(x, y) = xi-yj , (b) V(x,y) = yi-xj , (c) V(x, y, z) = xi + yi + zk x2+y2+z2 The DOT and CROSS PRODUCT 63. Find the projection of the vector 2i-3j + 6k on the vector i + 2j + 2k. Ans. 8/3 64. Find the projection of the vector 41-3J + k on the line passing through the points (2,3,-1) and (-2,-4,3). Ans. 1 65. If A = 4i-j + 3k and B =-2i + j-2k, find a unit vector perpendicular to both A and B. Ans. ±(i-2j-2k)/3 66. Find the acute angle formed by two diagonals of a cube. Ans. arc cos 1/3 or 70°326 7. Find a unit vector parallel to the xy plane and perpendicular to the vector 4i-3j +k. Ans. ± (3i +4j)/5 68. Show that A = (2i-2j +k)/3, B = (i +2j + 2k)/3 and C = (2i +j-2k)/3 are mutually orthogonal unit vectors. 69. Find the work done in moving an object along a straight line from (3,2,-1) to (2,-1,4) in a force field given by F = 41-3j+2k. Ans. 15 70. Let F be a constant vector force field. Show that the work done in moving an object around any closed polygon in this force field is zero. 71. Prove that an angle inscribed in a semicircle is a right angle. 72. Let ABCD be a parallelogram. Prove that AB2 + BC2 + CD2 +DA2 = AC2 + if 73. If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals, prove that AB2 + BC2 + CD-2 + DA2 = AC2 + YD-2 + 4 PQ2 This is a generalization of the preceding problem. 74. (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin. (b) Express the equation of (a) in rectangular coordinates. Ans. (a) r n = p , where n = A/A ; (b) A1x + A2 y + A3 z = Ap 75. Let r1 and r2 be unit vectors in the xy plane making angles a and R with the positive x-axis. (a) Prove that r 1= cos a i + sin a j, r2 = cos (3 i + sin I3 j. (b) By considering r1. r2 prove the trigonometric formulas cos (a-(3) = cos a cos a + sin a sin (3, cos ((% + S) = cos a cos(3-sin a sin R 76. Let a be the position vector of a given point (x1, y1, z1), and r the position vector of any point (x, y, z). Describe the locus of r if (a) I r-a I = 3, (b) (r-a). a = 0, (c) (r-a).r = 0. Ans. (a) Sphere, center at (x1, y1, z1) and radius 3. (b) Plane perpendicular to a and passing through its terminal point. (c) Sphere with center at (x1/2, y1/2, z1/2) and radius i xi+ y1+ z1, or a sphere with a as diameter.

Dialectica, 2009

Why (according to classical physics) do forces compose according to the parallelogram of forces? This question has been controversial; it is one episode in a longstanding, fundamental dispute regarding which facts are not to be explained dynamically. If the parallelogram law is explained statically, then the laws of statics are separate from and (in an important sense) "transcend" the laws of dynamics. Alternatively, if the parallelogram law is explained dynamically, then statical laws become mere corollaries to the dynamical laws. I shall attempt to trace the history of this controversy in order to identify what it would be for one or the other of these rival views to be correct. I shall argue that various familiar accounts of natural law (Lewis's Best System Account, laws as contingent relations among universals, and scientific essentialism) not only make it difficult to see what the point of this dispute could have been, but also improperly foreclose some serious scientific options. I will sketch an alternative account of laws (including what their necessity amounts to and what it would be for certain laws to "transcend" others) that helps us to understand what this dispute was all about.

This note briefly explains vectors suited for Cambridge AS and A-level mathematics, Cambridge IGCSE additional mathematics, and analysis and approaches mathematics for IB Diploma Programme.

The text is intended as some motivational survey of geometric algebra in 3D. Proofs are short and original. Multiplication of vectors is discussed in general, then basics of geometric algebra are founded. Among others, the special relativity, quantum mechanics and electromagnetic theories are discussed, operations on subspaces, dual, hyperbolic and complex numbers, linear transformations, idempotents, nilpotents, rotations, functions on multivectors, conformal model, calculus in geometric algebra, etc. Some problems are left to reader to solve.