Answer
$1.83$ rad
Work Step by Step
The formula to calculate the angle between two planes is:
$ \theta = \cos ^{-1} (\dfrac{p \cdot q}{|p||q|})$
Here, $p=\lt 1, \sqrt 2,-\sqrt 2 \gt$ and $q=\lt -1,1,1 \gt$
$|p|=\sqrt{(1)^2+(\sqrt 2)^2+(-\sqrt 2)^2+(0)^2}= \sqrt {5}$ and $|q|=\sqrt{(-1)^2+(1)^2+(1)^2}=\sqrt {3}$
Thus, $ \theta = \cos ^{-1} (\dfrac{p \cdot q}{|p||q|})=\cos ^{-1} \dfrac{-1}{ ( \sqrt {5})(\sqrt 3)})=\cos ^{-1} \dfrac{-1}{\sqrt {15}}$
or, $ \theta \approx 1.83$ rad
