Calculus-Vector Proofing

sufiya · New Member Offline Joined: 31 Oct 2009 Posts: 7

Given VectorOP=scalar(K)(one)*vectorOA+ScalaR(K)(two)*VectorOB and K(one)+k(two)=1
Prove that A,B,P are collinear (on the same line)

**Posted:** Sat Oct 31, 2009 5:01 am

Kent Merryfield

In $\LaTeX$ so we can all read it:

$\vec{OP} = k_1\vec{OA} + k_2\vec{OB},$ with $k_1 + k_2 = 1.$

This is linear algebra, of course, not calculus - but material like this does appear in calculus textbooks. This is also very much the standard way to parameterize a line.

One way to do this is to show that $\vec{AP}$ and $\vec{AB}$ are scalar multiples:

$\vec{AP} = \vec{OP} - \vec{OA} = (k_1\vec{OA} + k_2\vec{OB}) - \vec{OA}$

$= (k_1 - 1)\vec{OA} - k_2\vec{OB} = - k_2(\vec{OA} - \vec{OB}) = - k_2\vec{AB}.$

And we're done.

Note also that if $k_1$ and $k_2$ are both nonnegative (what is called a convex combination of $\vec{OA}$ and $\vec{OB}$ ), then $P$ lies on the line segment $\overline{AB}.$

**Posted:** Sat Oct 31, 2009 10:18 am