A unit vector making an obtuse angle with x-axis and perpendicular to the plane containing the points $\hat i+2\hat j+3k, 2\hat i+ 3\hat j+ 4\hat k$ and $\hat i+5\hat j+7\hat k$ also makes an obtuse angle with

z-axis

Let the given points be A, B and C respectively. The unit vectors perpendicular to the plane containing A, B
and C are given by

$±\frac{\vec{AB}×\vec{AC}}{|\vec{AB}×\vec{AC}|}=±\frac{1}{\sqrt{26}}(-\hat i+4\hat j-3\hat k)$

Since the required vector makes an obtuse angle with x-axis.

So, it is equal to $\frac{1}{\sqrt{26}}(-\hat i+4\hat j-3\hat k)$.

Clearly, it also makes an obtuse angle with z-axis.