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.