### Instruction: Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{c}\| = 2,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}.\]Find the smallest possible angle between $\mathbf{a}$ and $\mathbf{c},$ in degrees.### Response: This problem involves cross products of vectors, which are related to the angles between them.
The cross product $\mathbf{a} \times \mathbf{c}$ is a vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{c},$ and its magnitude is proportional to the product of their magnitudes and the sine of the angle between them.
Therefore, $\|\mathbf{a} \times \mathbf{c}\| = \|\mathbf{a}\| \|\mathbf{c}\| \sin \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{c}.$
The cross product $\mathbf{a} \times (\mathbf{a} \times \mathbf{c})$ is a vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{a} \times \mathbf{c},$ and its magnitude is proportional to the product of their magnitudes and the sine of the angle between them.
Since $\mathbf{a}$ and $\mathbf{a} \times \mathbf{c}$ are perpendicular, the angle between them is $\frac{\pi}{2},$ and the sine of that angle is $1.$
Therefore, $\|\mathbf{a} \times (\mathbf{a} \times \mathbf{c})\| = \|\mathbf{a}\| \|\mathbf{a} \times \mathbf{c}\|.$
Substituting the previous expression for $\|\mathbf{a} \times \mathbf{c}\|,$ we get $\|\mathbf{a} \times (\mathbf{a} \times \mathbf{c})\| = \|\mathbf{a}\|^2 \|\mathbf{c}\| \sin \theta = \|\mathbf{c}\| \sin \theta,$ since $\|\mathbf{a}\| = 1.$
The equation $\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}$ implies that $\mathbf{b}$ is the negative of $\mathbf{a} \times (\mathbf{a} \times \mathbf{c}),$ and therefore has the same magnitude.
So, $\|\mathbf{b}\| = \|\mathbf{a} \times (\mathbf{a} \times \mathbf{c})\| = \|\mathbf{c}\| \sin \theta.$
But $\|\mathbf{b}\| = 1$ and $\|\mathbf{c}\| = 2,$ so we can solve for $\sin \theta$ and get $\sin \theta = \frac{1}{2}.$
The smallest possible angle that has a sine of $\frac{1}{2}$ is $\frac{\pi}{6},$ which is $30$ degrees.
Therefore, the smallest possible angle between $\mathbf{a}$ and $\mathbf{c}$ is $30$ degrees.