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ps04sol

# ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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Problem Set 4 Solutions p. 1 of 7 Fall 2009 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2009 Problem Set 4 Solutions Problem 1: Two Vectors Given two vectors, ˆ ˆ ˆ (4 3 5 ) = + A i j k r and ˆ ˆ ˆ (7 4 4 = + + B i ) j k r , evaluate the following: (a) ; 2 + A B r r ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 2(4i 3j 5k) (7 4 4 ) 15 2 1 + = + + + + = + A B i 4 j k i j k r r (b) ; 3 A B r r 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 r r (c) ; A B r r Since and ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 = = = i i j j k k ˆ ˆ ˆ ˆ 0 = = = i j j k k i , the dot product is ( )( ) ( )( ) ( )( ) 4 7 3 4 5 4 36 = + − + = A B r r (d) ; × A B r r 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 r r 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 r r (e) What is the angle between and B A r r ? The dot product of and B is A r r cos θ = A B A B r r r r where θ is the angle between the two vectors. With: ( ) 2 2 2 (4) 3 (5) 50 5 2 A = = + − + = = A r 2 2 2 (7) (4) (4) 81 9 B = = + + = = B r , and using the result from part (c), we obtain 30 cos 0.6 52.5 . 5 2 9 θ θ = = = = A B A B r ° r r r

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