# Linear combination problem

1. Jan 14, 2009

### scarebyte

1. The problem statement, all variables and given/known data

Under what restrictions on c,d,e will the combinations c*u+d*v+e*w fill in the dashed triangle?

2. Relevant equations

3. The attempt at a solution

I think c,d,e>=0 and bcos the triangle it's in a plane maybe remove one of c,d or e doing c+d+e=number?

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2. Jan 19, 2009

### mutton

The midpoint between u and v is (u + v)/2. Convince yourself by drawing a parallelogram. This fact can be used to find conditions on c, d such that the linear combinations cu + dv fill in the line segment between u and v.

A better way to think about this: The vector from u to v is v - u. Starting at u, we can vary the length of v - u to fill in the line segment. Consider u + t(v - u) for t in [0, 1].

To fill in the entire triangle including its inside, use the same sort of reasoning: Starting at u, we can vary the lengths of v - u and w - u under certain restrictions.