Show that the line line is parallel to the plane . Also, find the distance between them.
Show that the line line is parallel to the plane . Also, find the distance between them.

Answer:

A line is parallel to the plane only when this line is perpendicular to the normal to the plane.

The line is parallel to the given plane.

Required distance between the line and the plane

\begin{array}{l}

=\frac{|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{n}}-\mathrm{q}|}{|\overrightarrow{\mathrm{n}}|}=\frac{|(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}) \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})-7|}{|\hat{\mathrm{i}}+\overline{\mathrm{j}}-\hat{\mathrm{k}}|} \\

=\frac{|2+5-7-7|}{\sqrt{1^{2}+1^{2}+(-1)^{2}}} \\

=\frac{|-7|}{\sqrt{3}}=\frac{7}{\sqrt{3}} \text { units. }

\end{array}