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# l_notes02 - Lecture II Vectors and Vector Algebra A set S...

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Unformatted text preview: Lecture II Vectors and Vector Algebra A set S of rays is called a direction if it satisﬁes the following laws: (1) Any two rays in S have the same direction. (2) Every ray that has the same direction as some member of S is in S . � A vector A consists of a non-negative real number, called the magnitude of � � the vector, and a direction. We denote the magnitude of A by |A|. � � A vector A such that |A| = 1 is called a unit vector. We will now describe four basic algebraic operations with vectors in E3 : 1 Multiplication by a scalar � � Let A by a vector, and let c by a real number. Multpying A by the scalar � c, we obtain a vector denoted by cA. The magnitude of the result is given � � � � by |cA| = |c||A|. The direction of cA is the same as the direction of A if � c ≥ 0, and opposite to the direction of A if c < 0. 2 Addition of vectors � � � � The sum of two vectors A and B is denoted simply by A + B . We can � deﬁne this sum geometrically. Translate B such that its start point is the � �� end-point of A. Then A + B will be the vector having the same start-point � � as A and the same end-point as B . 3 Scalar product (Dot product) � � The scalar product of two vectors A and B is a scalar quantity denoted �� �� �� � by A . B . Its value is A . B = |A||B |cosθ, where θ is the angle made by A � � � and B if we translate B such that it has the same start-point as A. We � � � observe that |B |cosθ is the length of the projection of B on A. 1 � B � A �� A+B Figure 1: Vector addition � The magnitude of A is equal to the square root of the dot product of A with itself: � | A| = Hence � A � |A| � = √A �� A .A � �� A .B is a unit vector with the same direction as A. 4 Vector product (Cross product) � � �� The cross product of two vectors A and B is a vector denoted by A × B . The magnitude of the cross product is given by: �� �� |A × B | = |A| |B |sinθ. � � Let A and B have the same start-point P and end-points Q1 and Q2 , � � �� respectively. Let M be the plane of A and B . The direction of A × B is normal to M in the manner established by the right-hand rule: if a right hand is placed at P and the ﬁngers are curling from P Q1 to T Q2 through �� the angle smaller than π , then the thumb indicates the direction of A × B . The following are some properties of these basic vector operations. � � � 1. (a + b)C = aC + bC � � 2. (ab)C = a(bC ) � � � � 3. a(B + C ) = aB + aC 2 �� � � 4. A + B = B + A � �� �� �� 5. A . (B + C ) = A . B + A . C � � �� 6. A × B = −B × A 3 ...
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## This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.

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