HW3-2 - ECE 493 Univ of Illinois HW#3 Version 1.00 Due Thur...

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ECE 493 HW #3 – Version 1.00 March 11, 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 . (b) Show that | -→ A · ( -→ B × -→ C ) | is the volume of parallelepiped generated by -→ A , -→ B and -→ C .
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