Draw a and b casually, a+b, that is, connect the A-tail with the B-head (translation). And tb, the unofficial vector B multiplied by any number, that is, the tail of B, can be located anywhere on the red line of the diagram?

The length of a+tb is the starting point of a, and the other point is taken on the red line. The minimum length of a+tb is the minimum distance from the point to the red line.

What is the angle between vector b and vector a+tb?

Draw it.

Oh, shit.

Isn't this vertical? It's only 90 degrees

If it weren't for the word vector, it would be a junior high school question.