Step 1: Use the definition of collinear vectors.
Two vectors are collinear (parallel) if and only if one is a scalar multiple of the other. So if $\vec{a}$ and $\vec{b}$ are collinear, then $\vec{b} = \lambda\,\vec{a}$ for some scalar $\lambda$. Here $\vec{a} = 2\hat{i}+3\hat{j}+6\hat{k}$, so $\vec{b} = \lambda(2\hat{i}+3\hat{j}+6\hat{k})$.
Step 2: Compute the magnitude of vector a.
\[ |\vec{a}| = \sqrt{2^2+3^2+6^2} = \sqrt{4+9+36} = \sqrt{49} = 7. \]
Step 3: Use the given magnitude of b to find lambda.
Since $\vec{b} = \lambda\,\vec{a}$, we have $|\vec{b}| = |\lambda||\vec{a}|$. Given $|\vec{b}| = 21$: \[ 21 = |\lambda| \times 7 \implies |\lambda| = 3. \] So $\lambda = 3$ or $\lambda = -3$ (both directions are valid since the problem does not specify orientation).
Step 4: Write out the vector b.
$\vec{b} = \pm 3(2\hat{i}+3\hat{j}+6\hat{k}) = \pm(6\hat{i}+9\hat{j}+18\hat{k})$.
Step 5: Verify the magnitude.
$|6\hat{i}+9\hat{j}+18\hat{k}| = \sqrt{36+81+324} = \sqrt{441} = 21$. Correct.
Step 6: State the final answer.
\[ \boxed{\vec{b} = \pm(6\hat{i}+9\hat{j}+18\hat{k})} \]