### Instruction: Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 6,$ $\|\mathbf{b}\| = 8,$ and $\|\mathbf{a} + \mathbf{b}\| = 11.$  Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$.### Response: I remember that the length of a vector sum is related to the lengths of the individual vectors and the angle between them by the law of cosines: $\|\mathbf{a} + \mathbf{b}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 - 2 \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta.$