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309-2008-Solutions1

# 309-2008-Solutions1 - ECE 3209 Electromagnetic Fields...

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University of Virginia Fall 2009 Homework # 1 Solutions 1. Cheng, 2-5. There are a number of ways to approach this problem. We are given that ~ A is a known vector and ~ X is an unknown vector. Also, the following products are known, p = ~ A · ~ X and ~ B = ~ A × ~ X To ﬁnd ~ X , we need to ﬁnd a basis — that is, a set of vectors that can be used to express ~ X . Since p is the projection of ~ A onto ~ X , ~ A is an easy choice for one of the basis vectors. Also note that the vector ~ B × ~ A is perpendicular to ~ A (and also to ~ B ). Thus we can use this as the second basis vector and write ~ X as, ~ X = a ~ A + b ( ~ B × ~ A ) where a and b are constants (scalars) that we need to ﬁnd. Consider the dot product given above, p = ~ A · ~ X = a ~ A · ~ A + b ~ A · ( ~ B × ~ A ) The last term above is zero (because ~ A is perpendicular to the vector ~ B × ~ A ), so noting that ~ A · ~ A = A 2 , a = p A 2 Now let’s consider the cross product, ~ B = ~ A × ~ X = a ~ A × ~ A + b ~ A × ( ~ B × ~ A ) Noting that ~ A × ~ A = 0 and using the vector identity, ~ A × ( ~ B × ~ C ) = ~ B ( ~ A · ~ C ) - ~ C ( ~ A · ~ B ) we have ~ B = ~ A × ~ X = b ~ B ( ~ A · ~ A ) - ~ A ( ~ A · ~ B ) Again the last term is zero because

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309-2008-Solutions1 - ECE 3209 Electromagnetic Fields...

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