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problem_set_5 - Name Section Student ID Last First MI Math...

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1 Name: First MI Last Student ID # Problem Set #5 Score Section: C REDIT 2 1 0 Problem #1 . Let I and J denote any two orthogonal unit vectors in R 3 which both happen to have a z –com- ponent of zero. Find, to within a sign, the value of I × J – and show all of your reasoning. Math 32A Lecture #5 Fall 2007 C REDIT 2 1 0 Problem #2 . Let A ( t ) and B ( t ) denote two vector valued functions of time. Derive a formula for the deriva- tive of the cross product: d dt A × B . Hint: You can try a derivation on the (primitive) definition which involves the basis vectors.
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2 Problem Set #5 C REDIT 2 1 0 Problem #4 . Let r A and r B denote (non–zero) vectors in R n . Although these have a dot product, you may not assume anything like the formula involving dot products and cosines of angles. In fact, something like this is what we will show in the form of an inequality. Part I. Show that for any scalar λ , | r A | 2 + λ 2 | r B | 2 2 λ r A r B . Part II. Find the optimal constant λ which makes the left and right side of the above inequality as close as pos- sible. Plugging in that value of λ , obtain a famous inequality involving | r A |, | r B | and r A r B .
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