app5 - Poularikas A.D Appendix 5 Vector Analysis The...

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Poularikas, A.D . Appendix 5 : Vector Analysis .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000
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© 2000 by CRC Press LLC Appendix 5: Vector Analysis Definitions Any quantity that is completely determined by its magnitude is called a scalar . Examples of such are mass, density, temperature, etc. Any quantity that is completely determined by its magnitude and direc- tion is called a vector . Examples of such are velocity, acceleration, force, etc. A vector quantity is represented by a directed line segment, the length of which represents the magnitude of the vector. A vector quantity is usually represented by a boldface letter such as V . Two vectors V 1 and V 2 are equal to one another if they have equal magnitudes and are acting in the same directions. A negative vector, written as – V , is one that acts in the opposite direction to V , but is of equal magnitude to it. If we represent the magnitude of V by υ , we write V = . A vector parallel to V , but equal to the reciprocal of its magnitude, is written as V –1 or 1/ V . The unit vector V / ( υ 0) is that vector which has the same direction as V , but which has a magnitude of unity (sometimes represented as V 0 or ). Vector Algebra The vector sum of V 1 and V 2 is represented by V 1 + V 2 . The vector sum of V 1 and – V 2 , or the difference of the vector V 2 from V 1 , is represented by V 1 V 2 . If r is a scalar, then r V = V r and represents a vector r times the magnitude of V , in the same direction as V if r is positive, and in the opposite direction if r is negative. If r and s are scalars and V 1 , V 2 V 3 vectors, then the following rules of scalars and vectors hold: V 1 + V 2 = V 2 V 1 ( r s ) V 1 r V 1 + s V 1 ; r ( V 1 + V 2 ) = r V 1 + r V 2 V 1 + ( V 2 + V 3 ) = ( V 1 + V 2 ) + V 3 = V 1 + V 2 + V 3 . Vectors in Space A plane is described by two distinct vectors V 1 and V 2 . Should these vectors not intersect one another, then one is displaced parallel to itself until they do ( Figure 1 ). Any other vector V lying in this plane is given by V r V 1 s V 2 . ˆ v
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© 2000 by CRC Press LLC A position vector specifies the position in space of a point relative to a fixed origin. If, therefore, V 1 and V 2 are the position vectors of the points A and B , relative to the origin O , then any point P on the line A B has a position vector V given by V = r V 1 + (1 – r ) V 2 . The scalar “ r ” can be taken as the parametric representation of P since r = 0 implies P = B and r = 1 implies P = A ( Figure 2 ). If P divides the line A B in the ratio r : s , then The vectors V 1 , V 2 V 3 , . .. , V n are said to be linearly dependent if there exist scalars r 1 r 2 r 3 , . .. , r n , not all zero, such that r 1 V 1 + r 2 V 2 L r n V n = 0 . A vector V is linearly dependent on the set of vectors V 1 V 2 V 3 , . .. , V n if V r 1 V 1 r 2 V 2 r 3 V 3 L + r n V n .
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This note was uploaded on 12/03/2009 for the course EEE transforn taught by Professor Profcenk during the Spring '09 term at Dogus University.

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app5 - Poularikas A.D Appendix 5 Vector Analysis The...

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