664HW1S_2010

664HW1S_2010 - SPRING 2010 ENME664: Solutions for...

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SPRING 2010 ENME664: Solutions for Assignment No. 1 1. Problems 1-3, 1-4, and 1-5 from Greenwood (1988) Problem 1-3 from Greenwood (1988) Given the following triad of unit vector: e 1 l 1 i l 2 j l 3 k e 2 m 1 i m 2 j m 3 k e 3 n 1 i n 2 j n 3 k where i , j , k are the usual Cartesian unit vectors. (a) Write 3 equations involving the ls, ms and ns which apply for any unit triad. (b) What additional equation applies if 12 3 , , and ee e are coplanar? (c) What 6 equations apply if 3 , , and e are mutually orthogonal? Solution : (a) 222 11 1 1 123 1 e e c o s lll   Similarly, m 1 2 m 2 2 m 3 2 1 and n 1 2 n 2 2 n 3 2 1 (b) The scalar triple product e ^ 1 ( e ^ 2 e ^ 3 ) 0 l 1 l 2 l 3 m 1 m 2 m 3 n 1 n 2 n 3 provides the additional equation that l’s, m’s and n’s should satisfy. (c) 1 2 2 2 3 3 90 0 o ee eec o s l mlm lm Similarly, l 1 n 1 l 2 n 2 l 3 n 3 0 and m 1 n 1 m 2 n 2 m 3 n 3 0 . The three equations obtained in part (a) along with the three equations shown above can be used as the 6 equations needed.
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Problem 1-4 from Greenwood (1988) Consider the vector A A x i A y j A z k . Find the component A 1 , A 2 , A 3 of the vector A in a skewed coordinate system whose axes have directions specified by the following unit vector triad: 12 3 ˆˆ () ˆ ˆ ; ; 22 ij ik ei e e   Solution : Expressing the unit vectors e ^ i in terms of i ^ , j ^ & k ^ , we obtain A A 1 i ^ A 2 ( i ^ j ^ ) 2 A 3 ( i ^ k ^ ) 2 (1) ( A 1 ( A 2 A 3 ) 2 ) i ^ A 2 2 j ^ A 3 2 k ^ (2)
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664HW1S_2010 - SPRING 2010 ENME664: Solutions for...

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