Vector making obtuse angle with x-axis

Question

Find unit vector making an obtuse angle with x-axis and perpendicular to the plane containing the points $A(1,2,3), B(2,3,4), C(1,5,7)$.

I managed to derive the plane containing the points $A, B, C$ viz. $x-4y+3z-2=0$ but cannot proceed from here.

Answer 1

Hint: for any plane given by the equation $ax + by + cz + d = 0$, the vector $(a, b, c)$ originating from the origin is perpendicular to the plane. So in your case $(1, -4, 3)$ is perpendicular to your plane. Now, it remains for you to answer these questions

• How do you modify that vector to get a unit vector?
• If the unit vector you got above is not obtuse with the $x$-axis, how do further modify it so that the new unit vector is? In fact, how do you determine the angles unit vectors make with the $x$-axis?