Exam 2 bonus (+10%) (from P. Yasskin and A. Belmonte)

Exam 2 bonus (+10%)
(from P. Yasskin and A. Belmonte)
Upon moving to a new city, you want to find an apartment which is
conveniently located relative to your school (point S in 2-dimensional space),
your place of work (point W ) and the shopping mall (point M ). You want to
minimize the sum of distances from your apartment A = (x, y) to the above
~ + |AW
~ | + |AM
~ |. Find and prove the geometric
three points: f (x, y) = |AS|
~ AW
~ and AM
~ which characterizes the
condition on the three vectors AS,
minimum of f . You may assume that A, W and M form an acute triangle.
~ expressing your answer in terms of the
1. Compute the gradient of |AS|
~ In particular, how are their directions related and what can
vector AS.
you say about the magnitude of the gradient?
2. Write the condition for the critical point in the vector notation.
3. The final answer is a condition on the angles between the vectors AS,
~ and AM
~ .
4. (hard, +10% extra) Discuss how the answer would change if one of
the angles in the triangle SW M were large (how large exactly?) and
why does the above proof fail (why differentiation fails to locate the
critical point).