### Instruction: 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.### Response: So the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.
That's right. We're given the magnitudes, so we just need to find the possible values of the cosine of the angle between them.
Right. The cosine of the angle between them can be anything from $-1$ to $1$, so the dot product can be anything from $-7 \cdot 11 = -77$ to $7 \cdot 11 = 77$.