Can three vectors in a plane in $\mathbb{R}^3$ be linearly independent? I believe the answer is no because in order to be on the plane they would have to be similar to the normal vector of the plane. For example I found a contradiction to this statement in the plane
$$z = x + y + w$$
with vectors:
$$[-1, 1, 1]$$
$$[1, -1, 0] $$
$$[0, 0, 1]$$
However I am unsure whether this applies to all vectors on the plane $\mathbb{R}^3$, although I am leaning more towards it is false.
A plane in $\mathbb{R^3}$ is a subspace of dimension 2 therefore you can't have more than 2 vectors in the plane linearly independent.
Another way to see that is consider that a plane is defined by one equation with tre variables
$$ax+by+cz=d$$
thus only two variables are free thus the dimension is 2.