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.
Best Answer
Hint: in a 3-dimensional space, $v_1, v_2, v_3$ being not coplanar is equivalent to saying that they are linearly independent, i.e., they form a basis. The question is to show that the same holds (or doesn't hold) for $w_1, w_2, w_3$.