l_notes03 - Lecture III Vector Algebra in Cartesian...

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Unformatted text preview: Lecture III Vector Algebra in Cartesian Coordinates Let us construct a Cartesian coordinates system in E3 . First we choose a point O, called the origin. Then we chose three mutually perpendicular rays starting from O. These rays are called the positive x axis, positive y axis, and positive z axis. Consider the lines containing these rays. For any of these lines, every point on it can be identified with a real number: if the point is on the ray, the real number is the distance to O, if it’s not on the ray, the number is the distance to O times −1. Let us denote these three lines by X, Y , and Z . Let P be a point in space. Consider the projection-points of P on X, Y , and Z . These points give the Cartesian coordinates of P , denoted xP , yP , and zP . Any triplet of real numbers forms the cordinatesfor some p oint P . Different points have different coordinates. The three unit vectors in the directions of the positive x, y , and z axes are ˆ � i j, customarily denoted by ˆ, ˆ and k . Let A be a vector in E3 and let P be the point � � such that OP = A . Let (a1 , a2 , a3 ) be the coordinates of P . Consider the vectors ˆ � � A1 = a1ˆ, A2 = a2 ˆ, and A3 = a3 k . by vector addition and multiplication with i� j scalars, one obtains the following expression: ˆ � � � � A = A1 + A2 + A3 = a1ˆ + a2 ˆ + a3 k i j � Then a1 , a2 , and a3 are called the scalar components of A, and a1ˆ a2 ˆ and i, j, ˆ are called the vector components of A. By using the ˆ ˆ k unit vectors, we ˆ � i, j, a3 k obtain coordinate formulas for the four basic vector operations: 1. Multiplication by a scalar ˆ � cA = (ca1 )ˆ + (ca2 )ˆ + (ca3 )k, i j for any scalar c. 1 2. Addition of vectors ˆ ˆ � � If A = a1ˆ + a2 ˆ + a3 k and B = b1ˆ + b2 ˆ + b3 k , then i j i j ˆ �� A + B = (a1 + b1 )ˆ + (a2 + b2 )ˆ + (a3 + b3 )k. i j 3. Dot product ˆ ˆ � � If A = a1ˆ + a2 ˆ + a3 k and B = b1ˆ + b2 ˆ + b3 k , considering that ˆ · ˆ = i j i j ii ˆˆ ˆ ˆi jˆ ˆj ˆ · ˆ = k · k = 1 and ˆ · ˆ = ˆ · ˆ = ˆ · k = k · ˆ = ˆ · k = k · ˆ = 0, we get that jj ij ji i �� A . B = a1 b1 + a2 b2 + a3 b3 . 4. Cross product ˆ ˆ. � � Let A = a1ˆ + a2 ˆ + a3 k and B = b1ˆ + b2 ˆ + b3 k To compute the cross i j i j product, we use the following equalities: ˆ × ˆ = ˆ × ˆ = k × k = 0, iijjˆˆ and ˆ j i, ˆ i ˆ × ˆ = −ˆ × ˆ = k, ˆ × k = −k × ˆ = ˆ k × ˆ = −ˆ × k = ˆ ij j i ˆj ˆ i ˆ j. We get the following formula: ˆ �� A × B = (a2 b3 − a3 b2 )ˆ + (a3 b1 − a1 b3 )ˆ + (a1 b2 − a2 b1 )k. i j Using determinants one can easily remember the formula, since: � � �ˆ ˆ k� ˆ� j �i � � �� A × B = � a1 a2 a3 � � � � � � b1 b2 b3 � � Considering that ˆ · ˆ = ˆ · ˆ = k · k = 1, we get that A can be expressed as ii jj ˆˆ �ii �ˆˆ � �jj A = (A.ˆ)ˆ + (A.ˆ)ˆ + (A. k )k, which is known as the frame identity. Triple products 1. Scalar triple product 2 ˆ� ˆ ˆ � � Let A = a1ˆ + a2 ˆ + a3 k , B = b1ˆ + b2 ˆ + b3 k ,and C = c1ˆ + c2 ˆ + c3 k . i j i j i j ��� The triple product (A × B ) . C is called a scalar triple product, since it is a scalar quantity. Its value is given by: � �a �1 � �� � × B ) . C = A . (B × C ) = � b1 �� � (A � � � c1 a2 b2 c2 � a3 � � � b3 � . � � c3 � ��� ��� Hence we can simply write (A×B ) . C as [A, B , C ] without specifying the positions for the cross and dot signs. We will use the following equalities in computing quadruple products: ��� ��� ��� ��� ��� ��� [A, B, C ] = [B, C, A] = [C, A, B ] = −[A, C, B ] = −[C, B, A] = −[B, A, C ]. 2. Vector triple product The cross product is not associative so we will give two formulas: �� � � �� ��� (A × B ) × C = (C . A)B −(C . B )A, and ��� ��� � � � A × (B × C ) = (A . C )B − (A . B )C Quadruple products 1. Scalar quadruple product ��� The expression (A × B ).(C × D) is called a quadruple scalar product, and by applying the formulas for triple products, we get the value: ��� � �� �� �� �� (A × B ).(C × D) = (C .A)(B .D) − (C .B )(A .D ) 2. Vector quadruple product � � � � The expression (A × B ) × (C × D) is called a quadruple vector product, and by applying the formulas for triple products, we get the value: �� � � � � �� �� �� (A × B ) × (C × D) = [C, D, A]B − [C, D, B ]A 3 ...
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