# Introduction to linear algebra (Strang) 1.1 problem 20

Under what restrictions on c, d, e, will the combinations c$$u$$+d$$v$$+e$$w$$ fill the dashed triangle? ($$u$$, $$v$$, $$w$$ are 3-d vectors)

I have been trying to see the way I could make restrictions on c, d and e, but the only thing I can think of is getting vectors from the vectors' heads and using the norm of the cross product, but this is a general case and I think it won't work for every case (not sure).

Then it just doesn't makes sense for me because it is supposed to be a linear combination that fills spaces not a plane, then if it had the characteristics to fill the plane, shouldn't it be a plane lying on the vectors and not intercepted by them?

• If you restrict to just $u$ and $v$, do you know for which scalars $a$ and $b$ that $au + bv$ lies in the line segment between $u$ and $v$? – Theo Bendit Mar 22 at 2:15