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# HW3-sol - ECE 493 HW#3 – Version 1.00 Spring 2011 Univ of...

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Unformatted text preview: ECE 493 HW #3 – Version 1.00 March 18, 2011 Spring 2011 Univ. of Illinois Due Thur, Mar. 17 Prof. Allen Topic of this homework: Rank-n-Span; Taylor series; Vector fields, Gradient Vector field topics Deliverables: Show your work. Numerical results are not sufficient, expect when specifically requested. General definitions: ˆ i , ˆ j , ˆ k are given in Greenberg on page 689: ˆ i · ˆ i = ˆ j · ˆ j = ˆ k · ˆ k = 1 , ˆ i · ˆ j = ˆ j · ˆ k = ˆ i · ˆ k = 0 , (1) ˆ i × ˆ i = 0 , ˆ i × ˆ j = ˆ k , ˆ k × ˆ j =- ˆ i , ··· . (2) 1. Triple product: Let-→ A = [ a 1 ,a 2 ,a 3 ] T ,-→ B = [ b 1 ,b 2 ,b 3 ] T ,-→ C = [ c 1 ,c 2 ,c 3 ] T be three vectors in R 3 . (a) Show that-→ A · (-→ B ×-→ C ) = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle . Solution: There are several ways of defining a vector that are used, and you need to learn to move between them. It is confusing to write a vector both as a column of three numbers, as-→ Z = [1 , 2 , 3] T , where the script T is transpose of the row-vector to a column, and as Z = ˆ i + 2 ˆ j + 3 ˆ k . They each represent the same thing. The arrow over the top indicates a column vector, but the rules of computation are the same. So you need to switch back and forth between the two ways of doing business....
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HW3-sol - ECE 493 HW#3 – Version 1.00 Spring 2011 Univ of...

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