
The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.
So, $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta$.

The maximum value of $\cos \theta$ is 1, which occurs when the vectors are pointing in the same direction.
So, the maximum value of $\mathbf{a} \cdot \mathbf{b}$ is $7 \cdot 11 \cdot 1 = 77$.

The minimum value of $\cos \theta$ is -1, which occurs when the vectors are pointing in opposite directions.
So, the minimum value of $\mathbf{a} \cdot \mathbf{b}$ is $7 \cdot 11 \cdot -1 = -77$.

Therefore, all possible values of $\mathbf{a} \cdot \mathbf{b}$ are in the interval $\boxed{[-77, 77]}$.The answer is: [-77,77]
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
If $\mathbf{a}$ and $\mathbf{b}$ are vectors such that $\|\mathbf{a}\| = 7$ and $\|\mathbf{b}\| = 11$, then find all possible values of $\mathbf{a} \cdot \mathbf{b}$.

Submit your answer in interval notation.

### Answer:
