I was doing a vector problem on Strang's Introduction to Linear Algebra. It asks me to find the restriction on c and d such that the vector $\vec{u} = c\vec{w} + d\vec{v}$, always ends at the dashed line.
My instinct tells me the relation between $c$ and $d$ is $c + d = 1$ and several examples confirmed my contention.
However, I find that I don't know how to prove this relation, that is, $c+d=1$. Any ideas? I'll appreciate any and all hints that help me get the ball rolling.
Observe that the vector $\vec{w}$ can be written as $\vec{v}+(\vec{w}-\vec{v})$. More generally, the vector $\vec{w}-\vec{v}$ points in the direction of the line of interest. Therefore, a vector of interest is of the form $\vec{v}+\lambda(\vec{w}-\vec{v})=\lambda\vec{w}+(1-\lambda)\vec{v}$. Now, add the coefficients to get one.