# Did I go about determining the coplanarity of these three vectors wrong?

I have a task stating this:

Determine if the following vectors are coplanar. Assume that $v_1$, $v_2$ and $v_3$ are not coplanar.

$w_1=4\vec v_1+3\vec v_2$

$w_2=\vec v_2+4\vec v_3$

$w_3=-\vec v_1-3\vec v_3$

I don't quite understand how I'll do this when I do not know the values of any of the vectors. Also, what significance does the information "$v_1$, $v_2$ and $v_3$ are not coplanar" have in terms of the solution? I'm guessing knowing that helps decide whether they're coplanar or not, but I can't see how.

In that question we arrived at the conclusion that they are coplanar. However, since then, I've attempted to find a way to mathmatically prove that they are coplanar, but instead I seem to arrive at the conclusion that they are in fact NOT coplanar. What I've done looks like this:

If the sum of the three vectors equals the zero vector in any way other than by multiplying every vector by zero, then they are coplanar. So if I can prove that one of the vectors can be represented as a multiple of the other, then they are coplanar. So to try and prove this, I’ll try to add two vectors.

$$\vec w_1+\vec w_2=(4\vec v_1+3\vec v_2)+(\vec v_2+4\vec v_3)$$

From there, I realized I could do this instead:

$$\vec w_1-3\vec w_2=(4\vec v_1+3\vec v_2)-3(\vec v_2+4\vec v_3)$$ $$\vec w_1-3\vec w_2=4\vec v_1+3\vec v_2-3\vec v_2-12\vec v_3$$ $$\vec w_1-3\vec w_2=4\vec v_1-12\vec v_3$$

Now the two vectors are written in the same terms as the third equation. However, this poses the problem that I need to something to multiply $(4-12)$ with to make it $(-1-3)$. Which simply isn't possible, meaning there is no way I can represent $\vec w_1$ and $\vec w_2$ as a linear combination of $\vec w_3$. This again ultimately means the vectors are linearly independent, and thus also not coplanar.

So, did I do something wrong here? Am I trying to prove this the wrong way, or was the answer for the other question simply wrong? As far as I can tell now, they are not coplanar, but in the other question we concluded that they were.