Normal to the plane $x+y+z=7$ is $\vec{n}=\hat{i}+\hat{j}+\hat{k}$
$\overrightarrow{A B}=-\hat{i}-\hat{k} \Rightarrow |\overrightarrow{A B}|=A B=\sqrt{2} $
$B C=\text { Length of projection of } \overrightarrow{A B} \text { on } \vec{n}=|\overrightarrow{A B} \cdot \hat{n}| $
$=\left|(-\hat{i}-\hat{k}) \cdot \frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\right|=\frac{2}{\sqrt{3}}$
Length of projection of the line segment on the plane is $A C$
$A C^{2}=A B^{2}-B C^{2}=2-\frac{4}{3}=\frac{2}{3}$
$A C^{2}=\sqrt{\frac{2}{3}}$
