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329sum09hw1sol - ECE-329 Summer 2009 Homework 1 Solution 1...

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ECE-329 Summer 2009 Homework 1 — Solution June 17, 2009 1. Given vectors A = 3ˆ x + ˆ y - z, B = ˆ x + ˆ y - ˆ z, C = ˆ x - y + 3ˆ z, we can calculate the following. a) The vector D = A + B = 4ˆ x + 2ˆ y - 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 | = x +5ˆ y - z 4 2 +5 2 +7 2 = x +5ˆ y - z 90 . e) The dot product A · B = (3ˆ x + ˆ y - z ) · 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 - y - z. 2. In a region, with electric field E and magnetic field B , the force experienced by a charged particle is F = q ( E + v × B ) , where q = 1 C . For three different particle velocities we get three different forces, so that z = E + 0 × B ˆ z = E + 1ˆ y × B z + 4ˆ y = E + 2ˆ z × B . The first equation directly gives E = 3ˆ z V m . Since B = B x ˆ x + B y ˆ y + B z ˆ z, we can rewrite the second and third equations as follows ˆ z = 3ˆ z - B x ˆ z + B z ˆ x z + 4ˆ y = 3ˆ z + 2 B x ˆ y - 2 B y ˆ x.
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