1.2 LECTURE 2. Scalar Product

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1 6 CHAPTER 1. VECTOR ALGEBRA Pythagean theem. cos 2 α 1 + cos 2 α 2 + cos 2 α 3 = 1 There is a one-to-one crespondence between the components of the vect on the one side and its magnitude and the direction angles on the other side. Position vect R = x i + y j + z k Displacement vect. The vect represented by the line segment betweeen the points P 1 = (x 1, y 1, z 2 ) and P 2 = (x 2, y 2, z 2 ) (with tail at P 1 and the tip at P 2 ) is R 2 R 1 = (x 2 x 1 ) i + (y 2 y 1 ) j + (z 2 z 1 ) k 1.2 LECTURE 2. Scalar Product Scalar Product First we prove that the cosine of the angle θ between any two nonzero vects A and B can be computed In Cartesian codinates by Proof: We compute A B 2 = A B cos θ = A, 0 θ π, B A B = A1 B 1 + A 2 B 2 + A 3 B 3 = (A i B i ) 2 = A 2 i + A i B i B 2 j 2 j=1 A k B k On the other hand, from the triangle fmed by the vects A, B, A B, we have A B 2 = ( B sin θ) 2 +( A B cos θ) 2 = A 2 + B 2 2 A B cos θ k=1 vecanal332.tex; August 24, 2017; 17:26; p. 6

2 1.2. LECTURE 2. SCALAR PRODUCT 7 This implies that A B = A k B k = A B cos θ k=1 This expression is called the scalar product ( the dot product) of the vects A and B. It is also called the inner product and denoted by ( A, B ). It is equal to the signed projection of one vect onto the other. The scalar square of a vect determines its nm A 2 = A A = A 2 Cauchy-Schwarz s Inequality. F any u, v E there holds The equality (u, v) u v. (u, v) = u v holds if and only if u and v are parallel. Maximum Principle. F a given vect A and a unit vect n the scalar product n A = A cos θ is maximal when n points in the same direction as A. Properties. F any vects A, B, C and scalars a, b A A 0, A A = 0 if and only if A = 0, A B = B A, 4. ( A + B ) C = A C + B C ), vecanal332.tex; August 24, 2017; 17:26; p. 7

3 8 CHAPTER 1. VECTOR ALGEBRA 5. (a A ) B = A (a B ) = a( A B ) F any u, v there holds u + v 2 = u 2 + 2(u, v) + v 2. ( A B ) 2 = A 2 + B 2 2 A B Triangle Inequality. F any u, v E there holds u + v u + v. Parallelogram Law. F any two vects u, v there holds u + v 2 + u v 2 = 2 ( u 2 + v 2) Therefe, the scalar product can be defined entirely in terms of the lengths of the vects 1 A B = { A 2 + B 2 A } B 2 2 Two non-zero vects u, v are thogonal, denoted by u v, if (u, v) = 0. A basis {e 1, e 2, e 3 } is called thonmal if each vect of the basis is a unit vect and any two distinct vects are thogonal to each other, that is, (e i, e j ) = δ i j, F an thonmal basis δ i j = { 1, if i = j 0, if i j A = A i e i, A i = e i A.. vecanal332.tex; August 24, 2017; 17:26; p. 8

4 1.2. LECTURE 2. SCALAR PRODUCT Lines and Planes A collection of vects fms a vect subspace if any linear combination of the vects from this set is a vect from this set,, alternatively, if any vect in this set can be written as a linear combination of vects from this set. Let A be a collection of vects. The span of A, denoted by span A, is the set of all finite linear combinations of vects from A of the fm v = a 1 e a k e k. We say that the subset span A is spanned by A. It is easy to see that the span of any subset of a vect space is a vect space. The span of a single vect u is the set of vects of the fm v = tu It defines a line passing through the igin in the direction of u. The span of a collection of two vects u and v is the set of vects of the fm v = tu + sv. If the vects u and v are nonparallel, then this defines a plane passing through the igin that is parallel to the vects u and v. If the vects u and v are parallel then they span a line parallel to one of the vects. Equation of a Line. Let P 0 = (x 0, y 0, z 0 ) be a fixed point and v be a given non-zero vect. Let P = (x, y, z) be a point on a line passing through the point P 0 and parallel to the vect v = a i + b j + c k. vecanal332.tex; August 24, 2017; 17:26; p. 9

5 10 CHAPTER 1. VECTOR ALGEBRA Then R R 0 = t v f some scalar t. Therefe, the parametric equation of the line is R = R 0 + t v t is a parameter that ranges in < t < (you may think of it as time). Since there is one parameter the line is a one-dimensional vect space spanned by v In components, the equation of the line is x = x 0 + at y = y 0 + bt z = z 0 + ct These gives 3 equations f 4 variables (x, y, z, t) leaving one parameter arbitrary. Alternatively, if we eliminate t (if a 0, b 0, c 0) we get the nonparametric equation of the line x x 0 a = y y 0 b = z z 0 c This gives two equations f 3 variables (x, y, z) leaving one arbitrary. Example. Find the equation of a line. Remarks. If a = 0 then the line is parallel to the yz-plane. The nonparametric equations of the line are x = x 0, y y 0 b = z z 0 c Example. Find the point of intersection of two lines R = 3 i + 2 j + (2 i + j + k )t vecanal332.tex; August 24, 2017; 17:26; p. 10

6 1.2. LECTURE 2. SCALAR PRODUCT 11 and x 3 2 = y 2 = z R = i 2 k + ( j + k )t Answer: P = (1, 1, 1). x = 1, y = z + 2, Equation of a Plane. A plane can be described by specifying a fixed point P 0 = (x 0, y 0, z 0 ) on the plane and two nonzero nonparallel vects A and B parallel to the plane. Let P = (x, y, z) be a point on the plane. Then the displacement vect R R 0 is parallel to the plane. Therefe, R = R 0 + t A + s B, t, s are scalar parameters. This is the parametric equation of the plane. Since there are two arbitrary real parameters, a plane is a two-dimensional vect space spanned by the vects A and B. To get the non-parametric equation of the plane we need to specify a vect N = a i + b j + c k, thogonal to the plane. Then it is thogonal to the displacement vect, therefe, N ( R R 0 ) = 0. a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 ax + by + cz = d d = ax 0 + by 0 + cz 0. This gives one equation f 3 variables (x, y, z) leaving two variables arbitrary. vecanal332.tex; August 24, 2017; 17:26; p. 11

7 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 vect N = a i +b j +c k. The equation of the plane passing through the point P 0 thogonal to the line L is a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = LECTURE 3. Vect Product Vect Product Let { e 1, e 2 } be an thonmal basis in the plane. We say that another basis { f 1, f 2 } is equivalent to the fmer basis ( has the same ientation) if it can be obtained from the first basis by a rotation and has the opposite ientation if it can be obtained from the first basis by a rotation and one reflection. We declare that the standard basis has positive ( right handed) ientation. Then there are just two types of bases, the ones with positive ientation (that have the same ientation as the standard one) and the one with negative ( left handed) ientation (that have the opposite ientation with the standard one). The ientation of the space is done similarly. We declare that the standard basis { i, j, k } has positive ( right-handed) ientation. Then all bases { e 1, e 2, e 3 } obtained by pure rotations from the standard one have positive ientation and those that need one reflection have negative ientation. Explain the right hand rule. The vect product ( cross product) of two non-zero vects, A, B, is defined by A B = A B sin θ n, vecanal332.tex; August 24, 2017; 17:26; p. 12