Regarding the regular triangular prism with edge length l
a) The vector representing the edge [B_1A_1] is vec v_1=(0,1,0)
and the vector representing the edge [BP] is vec v_2=(-sin(pi/3),cos(pi/3),1/2) Now making the dot product << vec v_1,vec v_2 >> = cos(pi/3) ne 0 so [B_1A_1] and [BP] are not perpendicular.
b) The vector associated to the edge [A A_1] is vec v_3=(0,0,1)
and the vector associated to [BP] is vec v_2=(-sin(pi/3),cos(pi/3),1/2) so their dot product is << vec v_3,vec v_2 >> = 1/2. We know that << vec v_3,vec v_2>> = norm (vec v_3)norm(vec v_2) cos(alpha) so cos(alpha)=(1/2)/( norm (vec v_3)norm(vec v_2) ) = (1/2)/(1cdot sqrt[5]/2) = sqrt(5)/5 ne sqrt(2)/2 so the angle between [A A_1] and [BP] is not pi/4
