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Problem Set 1 Solutions

Problem Set 1 Solutions - MASSACHUSETTS INSTITUTE OF...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2007 Problem Set 1 Solutions Problem 1: Two Vectors Given two vectors, ˆ ˆ ˆ (4 3 5 ) = + A i j k G and ˆ ˆ ˆ (7 4 4 ) = + + B i j k G , evaluate the following: (a) ; 2 + A B G G ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 2(4i 3j 5k) (7 4 4 ) 15 2 1 + = + + + + = + A B i G G 4 j k i j k (b) ; 3 A B G G ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 (4i 3j 5k) 3(7 4 4 ) 17 15 7 = + + + = − A B i j k i j k G G (c) ; A B G G Since and ˆ ˆ ˆ ˆ ˆ ˆ 1 = = = i i j j k k ˆ ˆ ˆ ˆ ˆ ˆ 0 = = = i j j k k i , the dot product is ( )( ) ( )( ) ( )( ) 4 7 3 4 5 4 36 = + − + = A B G G (d) ; × A B G G With ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ , and , × = × = × = i j k j k i k i j the cross product × A B G G is given by ˆ ˆ ˆ ˆ ˆ ˆ 4 3 5 32 19 37 7 4 4 × = = − + + i j k A B i j k G G (e) What is the angle between and A G B G ? The dot product of A and is G B G cos θ = A B A B G G G G where θ is the angle between the two vectors. With: ( ) 2 2 2 (4) 3 (5) 50 5 2 A = = + − + = = A G 2 2 2 (7) (4) (4) 81 9 B = = + + = = B G , and using the result from part (c), we obtain 30 cos 0.6 52.5 . 5 2 9 θ θ = = = = A B A B G ° G G G PS01 Solution - 1
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