# Ch1 - Ch.1 PHYS2004 Quantitative Methods for Basic Physics...

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Ch.1 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 1 Chapter 1 Vector Algebra, Coordinate Systems and R e l a t e d T o p i c s 1.1 Vector Algebra 1.1.1 Scalars and Vectors Scalars: quantities require only a real number to represent their size (or magnitude) e.g.: volume, mass, energy, speed, temperature, charge, … Vectors: quantities consist of a positive real number (modulus or magnitude) and a direction in space. e.g.: position, velocity, acceleration, force, electric field, … 1.1.2 Some Algebraic Rules for Scaling and Addition of Vectors: 1. If a and b are vectors, then ab is a vector defined by the parallelogram rule or the triangle rule. 2. Addition is commutative: abba   . 3. Addition is associative:    ab ca bc   . 4. If a is a vector and m is a scalar, then ma is a vector. 5. Scaling is associative in the sense that     12 1 2 mma . 6. Scaling is distributive in the sense that   1 2 mmam am a  . 7. Scaling is distributive over vector addition:   ma b m a m b . 8. The zero vector behaves as expected: 00 a and 0 aa .

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Ch.1 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 2 Addition of vectors Scaling of vectors Addition of vectors is associative. Scaling of vectors is distributive. 1.1.3 Products of Vectors The Dot/Scalar Product cos ab ab   , ( 1 . 1 . 1 ) where a and b are magnitudes of a and b respectively. The projection of vector b in the direction of a : a b a . In particular, ˆ ri x , ˆ rj y and ˆ rk z for ˆ ˆˆ rx iy jz k  .
Ch.1 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 3 The dot product of vectors v and u gives the projection vector p of v in the direction of u . Angle between a and b : b a b a c o s , ( 1 . 1 . 2 ) provided that neither a nor b is zero vector. Properties of the Dot Product 1. b a is a scalar 2. a b b a ( commutative ) 3. c a b a c b a ) ( ( distributive ) 4. ) ( ) ( ) ( b m a b a m b a m 5. 0 b a b a if both a and b are non-zero vectors. Given two vectors expressed in terms of the Cartesian unit base vectors, i.e. 12 3 ˆ ˆˆ aa ia ja k  and 3 ˆ bb ib jb k , 3 3 2 2 1 1 b a b a b a b a , ( 1 . 1 . 3 ) and the magnitude (norm) of a vector v is given by vv v  . The Cross/Vector Product  ˆ sin ab ab n  , ( 1 . 1 . 4 ) where ˆ n is a unit vector perpendicular to both a and b . The sign of ˆ n is given by the right hand screw rule .

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Ch.1 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 4 The direction of uv  determined by the right hand screw rule .
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## This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.

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Ch1 - Ch.1 PHYS2004 Quantitative Methods for Basic Physics...

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