It's not quite clear whether these are three separate questions, or one question with three consecutive rotations.
If these are three separate questions, you only have one free parameter in each case, the rotation angle, but a three-dimensional manifold of directions to cover, so it won't work.
If these are three consecutive rotations, the first one won't change the direction of the vector since it's parallel to the $x$ axis, but as Tony rightly pointed out, the remaining two rotations are enough to rotate it in the direction you want.
You can get the combined effect of rotations 2 and 3 by successively multiplying a vector $(1,0,0)$ along the positive $x$ axis by their rotation matrices:
$ \begin{eqnarray} \pmatrix{\cos\phi&-\sin\phi&0\\\sin\phi&\cos\phi&0\\0&0&1}\pmatrix{\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta}\pmatrix{1\\0\\0} &=& \pmatrix{\cos\phi&-\sin\phi&0\\\sin\phi&\cos\phi&0\\0&0&1}\pmatrix{\cos\theta\\0\\-\sin\theta} \\ &=&\pmatrix{\cos\phi\cos\theta\\\sin\phi\cos\theta\\-\sin\theta} \;. \end{eqnarray} $
You want this to be the unit vector along $(x,y,z)=(x_2,y_2,z_2)-(x_1,y_1,z_1)$, so you need
$\theta=-\arcsin\frac zr$
and
$\phi=\arctan\frac yx\;,$
where $r=\sqrt{x^2+y^2+z^2}$, and you'll want to use the atan2 function available in most programming environments to calculate the arctangent so you don't lose the sign information and don't have to divide by $0$ if $x=0$.