### Instruction: If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $43^\circ,$ what is the angle between the vectors $-\mathbf{a}$ and $\mathbf{b}$?### Response: To answer this question, I need to recall the geometric definition of the angle between two vectors.
If I draw the vectors $\mathbf{a}$ and $\mathbf{b}$ with their tails at the same point, the angle between them is the smaller angle that they form at their intersection.
If I draw the vector $-\mathbf{a}$ instead of $\mathbf{a},$ I am reflecting $\mathbf{a}$ across the origin, so it points in the opposite direction.
This means that the angle between $-\mathbf{a}$ and $\mathbf{b}$ is the supplementary angle of the angle between $\mathbf{a}$ and $\mathbf{b},$ since they add up to $180^\circ.$
Therefore, the angle between $-\mathbf{a}$ and $\mathbf{b}$ is $180^\circ - 43^\circ = 137^\circ.$