1.3 LECTURE 3. Vector Product

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1 12 CHAPTER 1. VECTOR ALGEBRA Example. Let L be a line x x 1 a = y y 1 b = z z 1 c passing through a point P 1 and parallel to the vector N = a i +b j +c k. The equation of the plane passing through the point P 0 orthogonal to the line L is a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = LECTURE 3. Vector Product Vector Product Let { e 1, e 2 } be an orthonormal basis in the plane. We say that another basis { f 1, f 2 } is equivalent to the former basis (or has the same orientation) if it can be obtained from the first basis by a rotation and has the opposite orientation if it can be obtained from the first basis by a rotation and one reflection. We declare that the standard basis has positive (or right handed) orientation. Then there are just two types of bases, the ones with positive orientation (that have the same orientation as the standard one) and the one with negative (or left handed) orientation (that have the opposite orientation with the standard one). The orientation of the space is done similarly. We declare that the standard basis { i, j, k } has positive (or right-handed) orientation. Then all bases { e 1, e 2, e 3 } obtained by pure rotations from the standard one have positive orientation and those that need one reflection have negative orientation. Explain the right hand rule. The vector product (or cross product) of two non-zero vectors, A, B, is defined by B = A B sin θ n, vecanal332.tex; August 25, 2017; 16:29; p. 12

2 1.3. LECTURE 3. VECTOR PRODUCT 13 where n is the unit vector orthogonal to both vectors A and B and such that the triple { A, B, n } is right-handed. The magnitude of the cross product is equal to the area of the parallelogram formed by A and B. Properties A = 0, B = B A, 3. ( A + B ) C = C + B C, 4. (a A ) B = (a B ) = a( B ) 5. A B = 0 if and only if the vectors are zero or parallel, The vector product of orthogonal unit vectors, e 1, e 2, is a unit vector e 3 = e 1 e 2 such that the triple { e 1, e 2, e 3 } is a right-handed orthonormal triple. In particular, if { e 1, e 2, e 3 } is an right-handed orthonormal basis then e 1 e 2 = e 3, e 2 e 3 = e 1, e 3 e 1 = e 2. This can be written in the form e i e j = ε ki j e k, i, j = 1, 2, 3, k=1 where ε i jk is the Levi-Civita symbol defined by ε 123 = ε 231 = ε 312 = 1 ε 213 = ε 321 = ε 132 = 1 and all other components are zero. vecanal332.tex; August 25, 2017; 16:29; p. 13

3 14 CHAPTER 1. VECTOR ALGEBRA This can be written in the form +1, if (i, j, k) is an even permutation of (1, 2, 3) ε i jk = 1, if (i, j, k) is an odd permutation of (1, 2, 3) 0, otherwise (1.3.1) The Levi-Civita symbol is completely anti-symmetric, that is, it changes sign under the permutation of any two indices, ε i jk = ε jik = ε k ji = ε ik j and does not change sign under the cyclic permutation of indices ε i jk = ε jki = ε ki j. Vector Product in Cartesian Coordinates. In an orthonormal basis A B = A i e i = i=1 i, j,k=1 B j e j j=1 ε ki j A i B j e k (1.3.2) That is, ( B ) k = ε ki j A i B j i, j This can be written in the form of a formal determinant i j k A B = ε i jk A j B k e i = A 1 A 2 A 3 B 1 B 2 B 3 Intersecting Planes. Consider two planes with the normals N 1 and N 2. If the vectors N 1 and N 2 are parallel then the plane are parallel and do not intersect. If the normals are not parallel, then the vector N 1 N 2 is non-zero and is parallel to the line L of the intersection of these planes. vecanal332.tex; August 25, 2017; 16:29; p. 14

4 1.3. LECTURE 3. VECTOR PRODUCT 15 Then the parametric equation of the line L passing through the point P 0 is R = R 0 + t N 1 N 2. Angular Velocity. Consider a rigid body rotating about a fixed axis with a constant angular speed ω. Then the velocity of the particle at the point R is v = ω R where ω is the angular velocity directed along the axis of rotation and with the magnitude ω = ω. The speed of the particle is v = ω R sin θ, where θ is the angle between R and the axis of rotation. Orthogonal Decomposition. Let A and B be two nonzero vectors. Let n = A A be the unit vector in the direction of A. Then the vector B can be decomposed as the sum of two orthogonal vectors B = B + B, where B = n ( n B ) = n B cos θ is the signed projection of B onto A and B = B B = ( n B ) n is the orthogonal component. vecanal332.tex; August 25, 2017; 16:29; p. 15

5 16 CHAPTER 1. VECTOR ALGEBRA Triple Product The triple product of three vectors A, B and C is defined by In tensor notation it is A ( B C ) = [ A, B, C ] = A ( B C ) i, j,k=1 ε i jk A i B j C k = A 1 A 2 A 3 B 1 B 2 B 3 C 1 C 2 C 3 It is equal to the signed volume of a parallelepiped based on three vectors A i, B j, C k Properties. 1. [A, B, C] = [B, C, A] = [C, A, B] = [B, A, C] = [C, B, A] = [A, C, B], 2. [A, B, C] = 0 if and only if the vectors are coplanar, 3. [A, B, C] is linear in each argument, 4. for an orthonormal basis [ e 1, e 2, e 3 ] = LECTURE 4. Tensors and Vector Identities We will denote the Cartesian coordinates by x 1 = x, x 2 = y, x 3 = z and the unit vectors in the direction of positive axes (called the standard basis vectors) by e 1 = i, e 2 = j, e 3 = k This can be denoted simply by x i and e j, where i, j = 1, 2, 3. For the indices one usually uses the lowercase Latin letters i, j, k, l, m, n etc. (do not confuse with i, j, k). If you run out of letters, you can use any other letters. The convention is though that the indices are denoted by small (versus capital) Latin (versus Greek) letters, and take values 1, 2, 3. Greek indices are used in four-dimensional space-time in special relativity, where they take values 0, 1, 2, 3, with x 0 = t denoting time. vecanal332.tex; August 25, 2017; 16:29; p. 16