Unformatted text preview: Contents
3 Vectors 3.1 Vectors and Scalars . . . . . . . . . . . . 3.1.1 Displacement vector . . . . . . . 3.1.2 Scalar multiplication on a vector 3.2 Adding Vectors Geometrically . . . . . . 3.3 Components of Vectors . . . . . . . . . . 3.4 Unit Vectors . . . . . . . . . . . . . . . . 3.5 Adding Vectors by Components . . . . . 3.6 Multiplying Vectors . . . . . . . . . . . . 3.6.1 Multiplying a vector by a scalar s 3.6.2 Scalar Product . . . . . . . . . . 3.6.3 Vector Product . . . . . . . . . . 3.6.4 Triple Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . 2 . 2 . 2 . 3 . 3 . 4 . 4 . 4 . 4 . 6 . 10 3
3.1 Vectors
Vectors and Scalars A scalar is a number which may be complex but is real if it represents physical quantity such as temperature, pressure, energy, mass and time. On the other hand, a vector has magnitude as well as direction. Geometrically, a vector can be described by an arrowed line pointing from a beginning point to an end point. In flat Euclidean space, two vectors are considered equal if they both have the same magnitude and point to the same direction. (For a general curved space in advanced geometry, vectors may depend where the beginning point resides.) A point particle moving in three dimensional space is specified by a vector that points from the origin of the coordinate system to the position of the particle in question. The vector is called the position vector x. Note in this course, a vector is always denoted by an arrow on top. The position vector of a moving particle is a function of time and is written as x (t). 1 3.1.1 Displacement vector If a particle changes its position by moving from A to B, we say that it  undergoes a displacement represented by the vector from A to B or AB. 3.1.2 Scalar multiplication on a vector For a vector a, its magnitude is often denoted by a or a . The multiplication of a scalar s to a vector a still gives us a vector sa. sa points in the same direction as a if s > 0 or in the opposite direction of a if s < 0. Furthermore, the magnitude sa = s a 3.2 Adding Vectors Geometrically
c=a+b The sum of vectors a and b is diagrammatically illustrated below: c b a Vector addition is commutative a+b=b+a and associative a+b +c=a+ b+c Define the subtraction a  b = a + (1) b 2 3.3 Components of Vectors A component of a vector is the projection of the vector on an axis. The projection of a vector on an x axis is its x component, the projection on the y axis is the y component and the projection on the z axis is the z component. z a y a3 a2 x a1 If vector a has a1 , a2 , a3 as the x, y, z components, then we also identity a = (a1 , a2 , a3 ) 3.4 Unit Vectors A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. The unit vectors in the positive x, y, and z direction ^ are labeled ^, , and k (or e1 , e2 , and e3 ). In terms of component notations, i ^ ^ ^ ^ we have ^ = e1 = (1, 0, 0) i ^ = e2 = (0, 1, 0) ^ ^ ^ ^ k = e3 = (0, 0, 1) Unit vectors are very useful for expressing other vectors; for example
3 ^ a = (a1 , a2 , a3 ) = a1^ + a2 + a3 k = i ^
i=1 ai ei ^ The length of a, expressed in terms its components, is a = a2 + a2 + a2 1 2 3 3 3.5 Adding Vectors by Components
3 3 Assume a = (a1 , a2 , a3 ) = ai ei , b = (b1 , b2 , b3 ) = ^
i=1 i=1 bi ei ^ Then
3 3 3 a+b =
i=1 ai ei ^ +
i=1 bi ei ^ =
i=1 (ai + bi ) ei ^ = (a1 + b1 , a2 + b2 , a3 + b3 ) 3.6
3.6.1 Multiplying Vectors
Multiplying a vector by a scalar s a = (a1 , a2 , a3 ) sa = (sa1 , sa2 , sa3 ) There are two ways to multiply a vector by another vector: one way to produce a scalar (called the scalar product, dot product, or inner product), and the other way to produce a new vector (called the vector product, cross product, or outer product). 3.6.2 Scalar Product The scalar or inner product of a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) is written as a b and defined to be
3 a b = a1 b1 + a2 b2 + a3 b3 = The scalar product is commutative ab=ba and distributive ai bi
i=1 (1) a b+c =ab+ac 4 The magnitude of a is given by a = For i = 1, 2, 3; j = 1, 2, 3; define ij , the Kronecker delta function as ij = Then and
3 aa 1, if i = j 0, if i = j ei ej = ij ^ ^
3 3 3 ab=
3 ai ei ^
i=1 3 bj ej ^
j=1 3 =
i=1 j=1 ai bj (^i ej ) e ^ =
i=1 j=1 ai bj ij =
i=1 ai bi By trigonometry, we have
2 2 2 ba = b sin2 +
2 b cos  a = a2 + b  2 a b cos where is the angle between a and b as shown in the following figure: b a But ba Thus, a b = a b cos The above expression instead of (1) can be taken as the definition of scalar product. Two vectors are orthogonal if their inner product vanishes. (zero vector is considered to be orthogonal to any vector.) 5
2 ba = b  a b  a = a2 + b  2a b 2 3.6.3 Vector Product The scalar product of a and b is written as a b and defined to be ab = e1 e2 e3 ^ ^ ^ a1 a2 a3 b1 b2 b3 = a2 a3 a a a a e1  1 3 e2 + det 1 2 e3 ^ ^ ^ b2 b3 b1 b3 b1 b2 = (a2 b3  a3 b2 ) e1 + (a3 b1  a1 b3 ) e2 + (a1 b2  a2 b1 ) e3 ^ ^ ^ Since a determinant changes sign under the interchange of any two rows e1 e2 e3 ^ ^ ^ a1 a2 a3 b1 b2 b3 e1 e2 e3 ^ ^ ^ =  b1 b2 b3 , a1 a2 a3 the vector product is anticommutative, a b = b a Now e1 ^ e2 ^ e3 ^ a1 a2 a3 b1 + c1 b2 + c2 b3 + c3 The vector is distributive a b+c =ab+ac Similarly, we also have a+b c= ac+bc = e1 e2 e3 ^ ^ ^ a1 a2 a3 b1 b2 b3 e1 e2 e3 ^ ^ ^ + a1 a2 a3 c1 c2 c3 6 vectors a, b, c. By definition ab c = = = So Triple Product The scalar a b c is called the triple product of three e1 e2 e3 ^ ^ ^ a1 a2 a3 b1 b2 b3 c a2 a3 a a a a c  1 3 c2 + det 1 2 c3 b2 b3 1 b1 b3 b1 b2 c1 c2 c3 a1 a2 a3 b1 b2 b3 = a1 a2 a3 b1 b2 b3 c1 c2 c3 = b1 b2 b3 c1 c2 c3 a1 a2 a3 a b c = b c a = (c a) b Since a1 a2 a3 a1 a2 a3 b1 b2 b3 b1 b2 b3 = 0, a1 a2 a3 b1 b2 b3 ab a= ab b=0 a b is perpendicular to a and b, and is thus perpendicular to the plane containing a and b. There are 9 possible cross products between two unit vectors ei and ej : ^ ^ e1 e1 ^ ^ e1 e2 ^ ^ e2 e3 ^ ^ e3 e1 ^ ^ = e2 e2 = e3 e3 = 0 ^ ^ ^ ^ = ^2 e1 = e3 e ^ ^ = ^3 e2 = e1 e ^ ^ = ^1 e3 = e2 e ^ ^ (2) For i, j, k {1, 2, 3}, the quantity ijk = (^i ej ) ek e ^ ^ is called the LeviCivita tensor. There are 33 = 27 possible i, j, k indices for ijk . ijk vanishes if any two of the three indices are identical. We are left 7 with 3! = 6 possible ijk with {i, j, k} = {1, 2, 3} that may be nonzero. In fact, from (2) we get 123 = 231 = 312 = 1 132 = 213 = 321 = 1 ijk = 0, otherwise ijk changes sign if any two indices are exchanged: ijk = jik = ikj = kji With ijk , it is straightforward to see that ab= ijk ai bj ek ^
i,j,k Now, a b is perpendicular to the plane that contains a and b. perpendicular to a b and must be in the plane containing a and b. Therefore a b c can be written as a linear combination of a and b. a b c = sa + tb a b c is which must also be perpendicular to c. Thus 0= So we may let s = ub c, t = ua c ab c=u b c a  (a c) b Furthermore, u must be a constant independent of the components of a, b, and c. This is because if we scale a sa, then a b c s a b c and u a sa. Similarly, u is unchanged under b sb or c sc. As a consequence, u is independent of the components of a, b, and c. Let us choose a = e1 , b = e2 ^ ^ and c = e1 . Then ^ (^1 e2 ) e1 = u ((^2 e1 ) e1  (^1 e1 ) e2 ) = u^2 e ^ ^ e ^ ^ e ^ ^ e 8 b c a  (a c) b su b c a  (a c) b . u is unchanged under a b c c = 0 = sa c + tb c So u = 1, and we have a b c =  b c a + (a c) b From the above identity, we get ab ab = a b a b = a2 b  a b a b
2 2 = a2 b  a b = a2 b sin2 2 where a b = a b cos . Thus the magnitude of a b is a b = a b sin Geometrical definition C B A The vector product of AB produces a vector whose magnitude is A B sin where is the smaller of the two angles between A and B. A B = A B sin ^ u In the above, the unit vector u, in the direction of A B, is perpendicular to ^ the plane that contains A and B, with the direction determined by the right hand rule: Sweep vector A into B with figures of your right hand and then your outstretched thumb shows the direction of A B. A BC = AC B AB C 9 3.6.4 Triple Product A B C is () the volume of the parallelepiped C B A with three sides formed by vectors A, B, and C. 10 ...
View
Full
Document
This note was uploaded on 05/14/2011 for the course ECON 101 taught by Professor Asdaf during the Spring '11 term at Universidad de San Buenaventura Bogota.
 Spring '11
 asdaf

Click to edit the document details