Homework 4 solutions

Andy Long

• Exercise 1: The easiest way to do this is via linear algebra. In particular,

  

  If we think of the and as components of vectors and , then

  

  or

  

  This is the inner product of two unit vectors, which is always between -1 and 1 (the inner product of two vectors is

  

  where is the angle between the vectors (and hence always between -1 and 1). We can demonstrate this fact using linear algebra, calculus (and a trick!) as follows:

  

  as this is the norm of the vector . Expanding this product as a function of t, we get

  

  Since this is always positive ( ), there are no roots: hence, the coefficients must satisfy the relationship

  

  This works out to

  

  or

  

  QED

• Exercise 2:

  

  so

  

  Thus

  

  If we can show that the middle term is zero, we're done.

  Recall that from our efforts to minimize we obtained two equations:

  

  and

  

  Now

  

  The second piece on the right hand side is clearly zero, as

  

  The second piece is also zero, but requires a little more work: from the second equation above, we have that

  

  Thus

  

  then, using the second equation,

  

  and so

  

  QED

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