Straight forward discrete trajectory toward a position in 2D, 3D, N-D

Dear all,

I'm trying to compute the trajectory of an object moving into a direction from a starting position 'pos' to a location 'loc'. This object is moving in a straight line. I am able to solve this in 2-D using sin and cos and multiple advices already posted here:

%Compute angular value in [0,2pi]:
alpha = atan2(loc(2) - pos(2), loc(1) - pos(1));
if(alpha < 0) alpha = alpha + 2 * pi; end

%Compute a step in the direction, with distance 'd':
pos(1) = pos(1) + d * cos(alpha);
pos(2) = pos(2) + d * sin(alpha);

However, in 3-D, i'm running in more trouble, I assume I should be using two angles and perform the step, ie:

%Get angular value in [0, 2pi]
alpha(1) = atan2(loc(2) - pos(2), loc(1) - pos(1));
alpha(2) = atan2(loc(3) - pos(3), loc(1) - pos(1));

if(any(alpha < 0)) %stay in [0, 2*pi]
    alpha(alpha < 0) = alpha(alpha < 0) + 2 * pi;
end

%Compute trajectory  
pos(1) = pos(1) + d * cos(alpha(1)) * cos(alpha(2));
pos(2) = pos(2) + d * sin(alpha(1)) * cos(alpha(2));
pos(3) = pos(3) + d * sin(alpha(2));

However, it doesn't work and I cannot figure out why? I assume there is something wrong with the angular values. I am also very interested in generalizing in N-D (N > 3) with a focus on both precision and efficiency.

Thank you