1. Translate the collinearity conditions into equations

Since $\overrightarrow a + 2\overrightarrow b$ is collinear with $\overrightarrow c$, there exists a non-zero scalar $m$ such that

Since $\overrightarrow b + 3\overrightarrow c$ is collinear with $\overrightarrow a$, there exists a non-zero scalar $n$ such that

We need to find

2. Use the fact that no two vectors are collinear

From (2),

So,

Substitute this into (1):

Rearrange:

Thus,

Now $\overrightarrow b$ and $\overrightarrow c$ are not collinear, so they are linearly independent in this relation. Therefore both coefficients must be zero:

Also,

Hence,

So,

3. Check options

• A: $\overrightarrow 0$ ✔️
• B: $\lambda \overrightarrow b$ ✘
• C: $\lambda \overrightarrow c$ ✘
• D: $\lambda \overrightarrow a$ ✘

So the correct option is A.

4. Comparison with stored correct answer

Stored correct answer: A

Derived answer: A

They agree.