### Instruction: If $\|\mathbf{a}\| = 3$ and $\|\mathbf{b}\| = 4,$ then find all $k$ for which the vectors $\mathbf{a} + k \mathbf{b}$ and $\mathbf{a} - k \mathbf{b}$ are orthogonal.### Response: I need to find all $k$ such that the dot product of $\mathbf{a} + k \mathbf{b}$ and $\mathbf{a} - k \mathbf{b}$ is zero.
To simplify the calculation, I can use the fact that $\|\mathbf{a}\| = 3$ and $\|\mathbf{b}\| = 4$ imply that $\mathbf{a} \cdot \mathbf{a} = 9$ and $\mathbf{b} \cdot \mathbf{b} = 16.$
So, the dot product is $(\mathbf{a} + k \mathbf{b}) \cdot (\mathbf{a} - k \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} - k \mathbf{a} \cdot \mathbf{b} + k \mathbf{b} \cdot \mathbf{a} - k^2 \mathbf{b} \cdot \mathbf{b} = 9 - k^2 16.$
This is zero when $k^2 = 9/16,$ so $k = \pm 3/4.$