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

The correct answer is Option (2) → z-axis

Unit vector 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 $\hat n=\frac{1}{\sqrt{26}}(-\hat i+4\hat j-3\hat k)$, consider a simple vector along z-axis, $\hat k=(0,0,1)$

$\hat n.\hat k=\frac{1}{\sqrt{26}}(0-0-3)=-\frac{3}{\sqrt{26}}$

Hence, the Angle between $\hat n$ and the z-axis is obtuse.