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Pset 1 Solutions

# Pset 1 Solutions - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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PS01-1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2010 Problem Set 1 Solutions Problem 1: Vectors (10 points) a) We need to use 33.7 A ! = ! in order to determine the x and y components of the vector A ! : cos (2.88)(cos(33.7 ) 2.40 x A A ! = = = A ! " , sin (2.88)(sin(33.7 ) 1.60 y A A ! = = = A ! " . Thus ˆ ˆ 2.40 1.60 = + A i j ! . We need to use 35.5 B ! = " ! in order to determine the x and y components of the vector B ! : cos (3.44)(cos( 35.5 ) 2.80 x B B ! = = " = B ! " , cos (3.44)(sin( 35.5 ) 2.00 y B B ! = = " = " B ! " . Thus ˆ ˆ 2.80 2.00 = ! B i j ! . b) The vector sum is then ˆ ˆ ˆ ˆ (2.40 1.60 ) (2.80 2.00 ) ˆ ˆ (5.20) ( .40) + = + + ! = + ! A B i j i j i j ! ! c) The vector difference is ˆ ˆ ˆ ˆ (2.40 1.60 ) (2.80 2.00 ) ˆ ˆ ( .40) (3.60) ! = + ! ! = ! + A B i j i j i j ! ! d) The unit vector pointing in the direction of A ! is given by ˆ ˆ 2.40 1.60 ˆ ˆ ˆ 0.83 0.69 2.88 = + = = = ! A A i j A i j A ! ! ! e) The unit vector pointing in the direction of B ! is given by ˆ ˆ 2.80 2.00 ˆ ˆ ˆ 0.81 0.58 3.44 ! = = = ! B i j B i j B ! ! Problem 2 Vectors (10 points) Consider the unit vector pointing from 1 r ! and 2 r ! given by 12 1 2 1 2 1 2 12 ˆ / / r = ! ! = ! r r r r r r r ! ! ! ! ! ! . The vector ! ! in the figure connects A ! to the point at 1 r ! , therefore we can write ! ! = x ˆ r 12 = x r 12 ! r 1 " ! r 2 ( ) . The vector ! r 1 = ! A + ! ! . Therefore, ! A = ! r 1 ! ! " = ! r 1 ! x r 12 ! r 1 ! ! r 2 ( ) = ! r 1 1 ! x r 12 # \$ % & ' ( + x r 12 ! r 2 .

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