# Calculus III Word Problem

• January 21st 2009, 09:02 PM
qbkr21
Calculus III Word Problem
I have absolutely no clue. Can someone at least get me started?

Thanks a Million!!!

http://i137.photobucket.com/albums/q...21/Eqn32-1.gif
• January 21st 2009, 10:21 PM
Vectors
Hello qbkr21
Quote:

Originally Posted by qbkr21
I have absolutely no clue. Can someone at least get me started?

Thanks a Million!!!

http://i137.photobucket.com/albums/q...21/Eqn32-1.gif

For the geometric argument, consider an arbitrary point in the plane, O, as origin. Then take two lines, $l$ and $m$, through O parallel to $\vec{a}$ and $\vec{b}$ respectively (effectively setting up a system of non-perpendicular* coordinate axes). Then consider point C whose displacement from O is represented by $\vec{c}$. From C, draw a line parallel to $l$, meeting $m$ at P. Then $\vec{PC} = s\vec{a}$ for some scalar $s$, and $\vec{OP} = t\vec{b}$ for some scalar $t$, and then:

$\vec{OC} = \vec{OP} + \vec{PC}$

$\Rightarrow \vec{c} = s\vec{a} + t\vec{b}$

For the components argument, let $\vec{a} = a_1\vec{i} + a_2\vec{j}$, and $\vec{b}$ and $\vec{c}$ similarly. Then consider separate components and show that equations

$c_1 = sa_1 + tb_1$

$c_2 = sa_2 + tb_2$

have a solution for some $s$ and $t$.

* PS. Not necessarily perpendicular is what I mean. Instead of being covered by a grid of unit squares, as in Cartesian geometry, the plane is spanned by parallelograms with sides $\vec{a}$ and $\vec{b}$.