how to find the two vectors, with known length, if summing them will align with another vector?

Question

I want to find 2 vectors, $v_x$ and $v_y$

with known length of each vector, $|v_x|$ and $|v_y|$ are both known

and summing them will align with another known vector $v_z$

which means $$v_x + v_y = kv_a$$ where k can be arbitrary scalar.

how to solve it?

I know there may be infinite possible solutions, so I think I will need a variable to control the final solution.

Thanks

Answer 1

Geometrically:

1. Pick any point $P$ on the line given by $v_a$ and inside the circle of radius $|v_x|+|v_y|$ around the origin $O$.
2. Draw the circle of radius $|v_x|$ around $O$ and the circle of radius $|v_y|$ arouns $P$
3. By choice of $P$, these two circles intersect. Let $Q$ be one of the points of intersection and let $v_x=\vec{OQ}$ and $v_b=\vec{QP}$.

Answer 2

Suppose you are given $\vec a, \vec b$ their combination should align with $\vec c$, then you requrie: $\vec a+ p\vec b= k \vec c~~~~(1)$ Dot this equation with $\vec a$ and then with $\vec b$ to get $$a^2+p (\vec a. \vec b)=k (\vec a. \vec c)~~~(2)$$ and $$\vec a. \vec b+p b^2=k (\vec b. \vec c)~~~(3)$$ These are two simple simultaneous equation,solve them to get $p,k$,