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Unformatted text preview: SPARK CHARTS Physics page 1 of 6 This downloadable PDF copyright 2004 by SparkNotes LLC. SPARK CHARTS TM PHYSICS SPARK CHARTS TM 50495 9781586636296 I SBN 1-58663-629-4 SPARK CHARTS Copyright 2002 by SparkNotes LLC. All rights reserved. SparkCharts is a registered trademark of SparkNotes LLC. A Barnes &amp; Noble Publication 10 9 8 7 6 5 4 3 2 Printed in the USA $4.95 $7.95 CAN SCALARS AND VECTORS A scalar quantity (such as mass or energy) can be fully described by a (signed) number with units. A vector quantity (such as force or velocity) must be described by a number (its magnitude) and direction. In this chart, vectors are bold: v ; scalars are italicized: v . VECTORS IN CARTE- SIAN COORDINATES The vectors i , j , and k are the unit vectors (vectors of length 1 ) in the x-, y-, and z-directions, respectively. In Cartesian coordiantes, a vector v can be writted as v = v x i + v y j + v z k , where v x i , v y j , and v z k are the components in the x-, y-, and z-directions, respectively. The magnitude (or length) of vector v is given by v = | v | = v 2 x + v 2 y + v 2 z . OPERATIONS ON VECTORS 1. Scalar multiplication : To multiply a vector by a scalar c (a real number), stretch its length by a factor of c . The vector v points in the direc- tion opposite to v . 2. Addition and subtraction: Add vectors head to tail as in the diagram. This is sometimes called the parallelogram method . To subtract v , add v . 3. Dot product (a.k.a. scalar product ): The dot product of two vectors gives a scalar quantity (a real number): a b = ab cos ; is the angle between the two vectors. If a and b are perpendicular, then a b = 0 . If a and b are parallel, then | a b | = ab . Component-wise calculation: a b = a x b x + a y b y + a z b z . 4. Cross product: The cross product a b of two vectors is a vector perpendicular to both of them with magnitude | a b | = ab sin . To find the direction of a b , use the right-hand rule: point the fingers of your right hand in the direction of a ; curl them toward b . Your thumb points in the direction of a b . Order matters: a b = b a . If a and b are parallel, then a b = 0 . If a and b are perpendicular, then | a b | = ab . Component-wise calculation: a b = ( a y b z a z b y ) i + ( a z b x a x b z ) j + ( a x b y a y b x ) k . This is the determinant of the 3 3 matrix a x a y a z b x b y b z i j k . Kinematics describes an objects motion. TERMS AND DEFINITIONS 1. Displacement is the change in position of an object. If an object moves from position s 1 to position s 2 , then the displacement is s = s 2 s 1 . It is a vector quantity....
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This note was uploaded on 01/08/2011 for the course ALL MISC taught by Professor Studyguides during the Spring '10 term at University of Florida.
- Spring '10