65_Vectors_in_R2_and_R3

65_Vectors_in_R2_and_R3 - Calculus and Vectors How to get...

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Calculus and Vectors – How to get an A+ 6.5 Vectors in R 2 and R 3 ©2010 Iulia & Teodoru Gugoiu - Page 1 of 4 6.5 Vectors in R 2 and R 3 A Polar Coordinates Given a Cartesian system of coordinates , a 2D vector v r may be defined by its magnitude || || v r and the counter-clockwise angle θ between the positive direction of the x-axis and the vector. The pair ) ||, (|| v r determines the polar coordinates of the 2D vector and ) ||, (|| v v r r = . Ex 1. Express each vector in polar coordinates in the form ) ||, (|| v v r r = . a) ) 45 , 6 ( 45 , 6 || || ° = ° = = a a r r b) ) 198 , 4 ( 198 72 270 , 4 || || ° = ° = ° ° = = b b r r B Scalar Components for a 2D Vector Let consider a 2D vector with the tail in the origin of the Cartesian system. Parallels through its tip to the coordinates axes intersect the x-axis at x v and the y-axis at x v . The pair ) , ( y x v v determines the scalar coordinates of the 2D vector and ) , ( y x v v v = r . Ex 2. Express each vector in scalar coordinates in the form ) , ( y x v v v = r . a) ) 3 , 2 ( = a r b) ) 1 , 4 ( = b r C Link between the Polar Coordinates and Scalar Components To convert a vector from the polar coordinates ) ||, (|| v v r r = to the scalar components ) , ( y x v v v = r , use the formulas: sin || || cos || || v v v v y x r r = = To convert a vector from the scalar components ) , ( y x v v v = r to the polar coordinates ) ||, (|| v v r r = , use the formulas: 2 2 || || y x v v v + = r (to get the magnitude) x y v v = tan (to get the direction) Ex 3. Do the required conversions. a) Convert ) 120 , 10 ( ° = a r to the scalar coordinates. ) 3 5 , 5 ( 3 5 120 sin 10 sin || || 5 120 cos 10 cos || || 120 , 10 || || = = ° = = = ° = = ° = = a a a a a a y x r r r r b) Convert ) 7 , 4 ( = b r to the polar coordinates. ) 26
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65_Vectors_in_R2_and_R3 - Calculus and Vectors How to get...

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