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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.

Ask a homework question - tutors are online