Answer
$cos \theta =\frac{6}{\sqrt {42}}$
or, $\theta =22.21 ^\circ$
Work Step by Step
Given: Planes $x+y+z=0$ and $x+2y+3z=1$
$n_1=\lt 1,1,1\gt$
and
$n_2=\lt 1,2,3\gt$
To find the cosine of the angle, use formula:
$cos \theta =\frac{n_1 \cdot n_2}{|n_1| |n_2|}$
$cos \theta =\dfrac{\lt 1,1,1\gt \cdot \lt 1,2,3\gt}{\sqrt {1^2+1^2+1^2}{\sqrt {1^2+2^2+3^2}}}$
$cos \theta =\frac{6}{\sqrt {42}}$
or, $\theta =22.21 ^\circ$