I have the problem:
Decompose the vector $\vec v = (1,2,4)$ in two parts, one parallel to the plane $X = (1,1,0) + \lambda(1,0,1) + \gamma(0,1,-1)$ and other parallel to the line $X = (0,0,0) + h(2,1,0)$
What i've done:
I tried to project the vector $\vec v$ in the direction of $(1,0,1)$ and decompose it in the direction of $(0,1,-1)$, then the sum of its two projections is parallel to the plane. I, then, decomposed $\vec v$ in the direction of $(2,1,0)$. But my answer is wrong. Does it has something to do with projections or I'm doing the wrong way? i checked my projections in wolfram alpha and they seemed good.