I need help with problem 3 2. Let u and v be two vectors in Ra. Let a luli filu.

Question

I need help with problem 3

2. Let u and v be two vectors in Ra. Let a luli filu. Prove thBt 8 bisects the angle between u and V 3. A box with square sides has a valume V. he coet of making the top is $a per unit the sides cost $b per unit and the bottom Sc par unit a) If the volume s fixed find the dimensipris lhat wi il minimize the cost b) lf the cost is fixed find the dimensions that will maximize the volume. c) Show that the two solutions are the same. 4. If a 0 and ax bx c has no real faots, find bx

Explanation / Answer

Solution:

1)Solution. Let ~w = ||~v||~u + ||~u||~v.

Let 1 be the angle between the vector ~u and ~w.

Let 2 be the angle between the vector ~v and ~w.

By the property of the dot product, we have cos(1) = ~u· ~w ||~u|| || ~w|| and cos(2) = ~v· ~w ||~v|| || ~w||. Note tha

t u · w = u · (||v||u + ||u||v) = ||v||u · u + ||u||u · v

. Thus cos(1) = ||v||·||u||2+||u||u·v /||~u|| || ~w|| = ||~v|| ||~u||+~u·~v || ~w|| and

cos(2) = ||v|| v·u+||u|| ||~v||2/||v||·|| w|| = v·u+||~u|| ||~v|| || ~w|| .

Since v · u = u · v and ||v|| ||u|| = ||u|| ||v||, we have cos(1) = cos(2).

This implies that the angle between u and w = ||v||u + ||u||v is the same as the angle between ~v and ~w = ||~v||~u + ||~u||~v. Thus ||v||u + ||u||v bisects the angle between u and v.

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