Collinearity of 3 vectors

Question

a,b,c are three non-coplanar vectors. The points P,Q,R with position vectors:

$$\mathrm P: a-2b+3c; \\ \mathrm Q: 2a+\mathrm Xb-4c \\ \mathrm R: -7b+10c$$

will be collinear if the value of "X" is?

Answer 1

The three vectors are coplanar iff they are linearly dependent, and this means that: $$ \det \begin{bmatrix} 1&-2&3\\ 2&X&-4\\ 0&-7&10 \end{bmatrix} = 0 $$

Answer 2

Three points $P, Q, R$ are collinear if vectors $PQ$ and $PR$ "point to the same direction". That means $PQ = x * PR$. I suggest to calculate $PQ$ and $PR$ in terms of $a, b, c$ and check if they can "point to the same direction" for some $X$.