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p2sol - Physics 21 Problem Set 2 Solutions Richard Eager...

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Physics 21 Problem Set 2 – Solutions Richard Eager Department of Physics University of California; Santa Barbara, CA 93106 Problem 1 (KK 1.8) Given vectors A = ˆ i + ˆ j - ˆ k B = 2 ˆ i - ˆ j + 3 ˆ k we compute their cross product as follows, A × B = ˆ i ˆ j ˆ k 1 1 - 1 2 - 1 3 using cofactor expansion = 1 - 1 - 1 3 ˆ i - 1 - 1 2 3 ˆ j + 1 1 2 - 1 ˆ k = 2 ˆ i - 5 ˆ j - 3 ˆ k A unit normal perpendicular to A and B is A × B A × B = 2 ˆ i - 5 ˆ j - 3 ˆ k 38 Problem 2 (KK 1.11) We use the following lemma, proved in section: Lemma 0.1. Either A B or A, B, and A × B form a basis. We apply this lemma to A, ˆ n, and A × ˆ n. It is easy to show A = ( A · ˆ n n + (ˆ n × A ) × ˆ n if A and ˆ n are parallel. So we treat the di ffi cult case in which they aren’t parallel, but from the lemma, form a basis once we include A × ˆ n. We will show that the equality holds in each of the three components formed by our basis, so as a consequence the equation is true as a vector equation. In the A direction, A 2 = ( A · ˆ n ) 2 + (ˆ n × A ) 2 = A 2 cos 2 θ + A 2 sin 2 θ = A 2 1
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In the ˆ n direction, A · ˆ n = ( A · ˆ n n 2 + ((ˆ n × A ) × ˆ n ) · ˆ n since (ˆ n × A ) × ˆ n is perpendicular to ˆ n = A · ˆ n In the A × ˆ n direction, A · ( A × ˆ n ) = 0 , so we must show ( A · ˆ n n + (ˆ n × A ) × ˆ n · ( A × ˆ n ) = 0 .
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