Under which of the following conditions will the points $A, B, C$ with position vectors $\vec a$, $\vec b$ and $\vec c$ respectively be collinear?
(a) $\vec c-\vec a = 2(\vec b-\vec a)$
(b) $|\vec c-\vec a|=2|\vec b-\vec a|$
(c) $\vec a=2(\vec b+\vec c)$
(d) $2\vec a+\vec b=\vec c$
(e) $3\vec a-2(\vec b+\vec c)=\vec 0$
My Try
Since $(a)$ yields $\vec {AC}=\lambda \vec {AB}$, it is definitely collinear.
Furthermore I know that magnitude does not affect collinearity. I'm stuck determining $(c), (d), (e)$. Please help me. Thanks in advance!
Note that the collinearity $\vec {AC}=\lambda \vec {AB}$ is equivalent to the vector equation $$ \vec c = \lambda \vec b + (1-\lambda )\vec a$$
where $\lambda $ is non-zero. Then, verify that $(c), (d), (e)$ do not satisfy the equation above.