If you drag around point B, you will see the green angle change. This vector AB is the given vector, and the green angle is the direction angle of that vector. The red angle and the blue angle are the other two direction angles that will provide you with vectors orthogonal to vector AB. As you drag around point B, you will notice that the difference between the green and red angle, or green and blue angle is always 90 degrees.
So, in order to get the red angle, we always end up doing the green direction angle minus 90 degrees. In order to get the blue angle, we always end up doing the given green angle plug 90 degrees. The blue and red direction angles are the ones we can use to write a vector that is orthogonal to the given vector.