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Unformatted text preview: ECE329 Spring 2009 Homework 1 — Solution January 26, 2009 1. Given vectors A = 3ˆ x + ˆ y 2ˆ z, B = ˆ x + ˆ y ˆ z, C = ˆ x 2ˆ y + 3ˆ z, we can calculate the following. a) The vector D = A + B = 4ˆ x + 2ˆ y 3ˆ z. b) The vector A + B 4 C = 10ˆ y 15ˆ z. c) The vector magnitude  A + B 4 C  = √ 10 2 + 15 2 = √ 325 . d) The unit vector ˆ u = A +2 B C  A +2 B C  = 4ˆ x +5ˆ y 7ˆ z √ 4 2 +5 2 +7 2 = 4ˆ x +5ˆ y 7ˆ z √ 90 . e) The dot product A · B = (3ˆ x + ˆ y 2ˆ z ) · (ˆ x + ˆ y ˆ z ) = 3 + 1 + 2 = 6 . f) The cross product B × C = ˆ x ˆ y ˆ z 1 1 1 1 2 3 = (3 2)ˆ x (3 + 1)ˆ y + ( 2 1)ˆ z = ˆ x 4ˆ y 3ˆ z. g) The scalar triple product A · ( B × C ) = (3ˆ x + ˆ y 2ˆ z ) · (ˆ x 4ˆ y 3ˆ z ) = 3 4 + 6 = 5 . h) If vectors A , B , and C have units of meters, then the units of A · ( B × C ) would be m 3 .  A · ( B × C )  is the volume of a parallelepiped formed by these vectors. 2. Volume integral review. Given the charge density ρ ( x,y,z ) = x 2 + y 2 + z 2 C / m 3 , the total electric charge contained in a cube of volume V = 8 m 3 centered at the origin is (1 , 1 , 1) ( 1 , 1 , 1) ( 1 , 1 , 1) (1 , 1 ,...
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This note was uploaded on 11/16/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 FRANKE
 Electromagnet

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