a) No problem, I have done this

b) I found the orthogonal basis of B to be {{1,0,1}{1/2,1,-1/2}} - not sure if correct, but I have done it several times and come up with the same answer each time

c) This one I am totally unsure about. I found the shortest distance to be sqrt(3)/3

d) The cross product of 1,0,1 and 1/2, 1, -1/2 is -1, 1, 1 so the other vector is 1, -1, -1 (perpendicular in opposite direction)

a and d I am confident about, b and c, totally unsure, but not sure how to go about checking? Is someone able to independent confirm?