w8-03-web

# w8-03-web - hapter 3 Vectors “Vectors and Scalars” says...

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Unformatted text preview: hapter 3 Vectors “Vectors and Scalars” says Jearl. How about “scalars and vectors and tensors ” instead? No reason why not! Scalar: temperature, T = 30 C ˚ Vector: velocity r v hapter 3 Vectors A B C AC is the vector sum of the vectors AB and BC r a r b r s “sum” is here a new use of the word - it is NOT the algebraic sum! 'Why did you call him Tortoise, if he wasn't one?' Alice asked. 'We called him Tortoise because he taught us,' said the Mock Turtle angrily. 'Really you are very dull!' 'I only took the regular course.' 'What was that?' inquired Alice. 'Reeling and Writhing, of course, to begin with,' the Mock Turtle replied; ' and then the different branches of Arithmetic--Ambition, Distraction, Uglification, and Derision.' 'I never heard of "Uglification",' Alice ventured to say. 'What is it?' hapter 3 Vectors hapter 3 Vectors A C is the vector sum of the vectors and r a r b r s r b r a r s r b r a hapter 3 Vectors A C r a r b r s r b r a r s = ρ α+ ρ β r s = ρ β + ρ α r b + ρ α = ρ α+ ρ β “Commutative Law” says Jearl. Klein, “Mathematical Thought...” “By the middle of the nineteenth century the axioms of algebra generally accepted were: 1.) Equal quantities added to a third yield equal quantities. 2.) (a + b) + c = a + (b + c) 3.) a + b = b + a 4.) Equals added to equals give equals. 5.) Equals added to unequals give unequals. 6.) a(bc) = (ab)c 7.) ab = ba 8.) a(b + c) = ab + ac Two-dimensional vectors obey all these rules; three-dimensional vectors do not! 7.) ab = ba hapter 3 Vectors You can move vectors around! Vectors have MAGNITUDE and DIRECTION but they do NOT have POSITION a vector You can move vectors around! Components of Vectors This is where it gets easy! Everything just turns into High School algebra! x y Introduce a coordinate system x y x y r a a x a y Vector has components a x , a y ( 29 r a θ x y α x y α x y Vector r a a x a y has components a x , a y ( 29 r a θ Why does our vector have TWO components? Remember “scalars and vectors and tensors ” ? A scalar, e.g. temperature, has ONE component A vector, e.g. velocity, has ... three components? (If space is 3- d, e.g. a room) v x , v y , v z ( 29 velocity, has ... four components? (If spacetime is 4-d, which it is!) v x , v y , v z , v t ( 29 velocity , has ... two compon ents? space is 2-d, e.g. a blackbo ard) v x , v y ( 29 Vectors have ONE subscript, which has as many values as the space you are working in has Dimensions. velocity, has ... four components? (If spacetime is 4-d, which it is!) v x , v y , v z , v t ( 29 But might there be measured quantities that have TWO subscripts, not one? Or THREE subscripts? Or FOUR, or FIVE? But might there be measured quantities that have TWO subscripts, not one?...
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w8-03-web - hapter 3 Vectors “Vectors and Scalars” says...

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