Okay, let's see. I need to find the smallest possible angle between vectors a and b given their magnitudes and their cross product. Hmm. Alright, let's start by recalling some vector properties. 

First, I remember that the magnitude of the cross product of two vectors a and b is equal to the product of their magnitudes times the sine of the angle between them. The formula is |a × b| = |a||b|sinθ. So, if I can find the magnitude of the cross product, I can solve for sinθ, and then find θ.

Given that a × b is [3, 2, 6], the magnitude of this cross product is the square root of (3² + 2² + 6²). Let me calculate that. 3 squared is 9, 2 squared is 4, and 6 squared is 36. Adding those up: 9 + 4 + 36 = 49. The square root of 49 is 7. So |a × b| = 7.

Wait, but the cross product magnitude formula is |a||b|sinθ. We know that |a| is 2 and |b| is 7, so plugging those in: 2 * 7 * sinθ = 14 sinθ. And we just found that this equals 7. So 14 sinθ = 7. Therefore, sinθ = 7 / 14 = 0.5. 

So sinθ = 0.5. The angle θ whose sine is 0.5. The principal value for that is 30 degrees, since sin(30°) = 0.5. But wait, sine is also positive in the second quadrant, so 150 degrees. But since we are asked for the smallest possible angle between them, that would be 30 degrees. So is that the answer?

Wait, but hold on. The cross product's magnitude is 7, which gives sinθ = 0.5. But the angle between two vectors is defined between 0 and 180 degrees, right? So the angle θ can be either 30 degrees or 150 degrees. The problem asks for the smallest possible angle. So 30 degrees is smaller than 150 degrees, so the answer should be 30 degrees.

But wait, is there a catch here? Because cross product's direction is also important. The cross product vector's direction is perpendicular to both a and b, following the right-hand rule. But the magnitude only depends on the sine of the angle, regardless of direction. So the magnitude of the cross product gives us |a||b|sinθ, but the actual direction of the cross product vector is determined by the right-hand rule. However, for the purpose of finding the angle between a and b, the angle is the smallest one between them, which is between 0 and 180 degrees. So regardless of the direction of the cross product, the angle is determined by the sine value.

Therefore, sinθ = 0.5 gives θ = 30 degrees or 150 degrees, and the smallest one is 30 degrees. So the answer should be 30 degrees.

But wait, let me verify. Let's double-check the calculations. The cross product vector is [3, 2, 6], so its magnitude is sqrt(3² + 2² + 6²) = sqrt(9 + 4 + 36) = sqrt(49) = 7. Correct. Then |a||b|sinθ = 2*7*sinθ = 14 sinθ. Therefore, 14 sinθ = 7 → sinθ = 0.5. So θ is 30 degrees or 150 degrees. Since we want the smallest angle, 30 degrees is the answer. That seems right.

But hold on, is there a scenario where even though sinθ is 0.5, the actual angle could be something else? For instance, if the vectors are not in the same plane or something? Wait, no. The angle between two vectors is always between 0 and 180 degrees, regardless of their orientation in space. So the cross product's magnitude only depends on the sine of the angle between them. Therefore, regardless of the cross product's direction, the magnitude gives us sinθ, and the angle is the smallest one. So 30 degrees is the answer.

Therefore, I think the answer is 30 degrees. Let me just make sure there's no alternative approach here. For example, sometimes when dealing with cross products, there might be cases where the vectors are constrained in some way, but in this case, we're given the cross product vector, so we can directly compute its magnitude and use that. So I think that's solid.

Another way to think about it: the cross product's magnitude is the area of the parallelogram formed by the vectors a and b. So if the area is 7, and the product of the lengths is 14, then the sine of the angle is 0.5, leading to the same conclusion. So yes, 30 degrees is the smallest angle between them. 

Therefore, after all this thinking, I believe the answer is 30 degrees.
Therefore, the smallest possible angle between $\mathbf{a}$ and $\mathbf{b}$ is $\boxed{30}$ degrees.
</think>