State when the line ` vec rdot vec a+lambda vec b` is parallel to the plane ` vec rdot vec n=ddot` Show that the line ` vec r= hat i+ hat j+lambda(2 hat i+ hat j+4 hat k)` is parallel to the plane ` vec rdot(-2 hat i+ hat k)=5.` Also, find the distance between the line and the plane.

Explanation

A line is perpendicular to a plane when the normal vector of the plane is perpendicular to the line.


`\Rightarrow \vec{b} \cdot \vec{n}=0`


`\vec{a}=\hat{i}+\hat{j} \quad \vec{b}=2 \hat{i}+\hat{j}+4 \hat{k} \cdot \vec{n}=-2 \hat{i}+\hat{k}`.



`\vec{b} \cdot \vec{n}=(2 \hat{i}+\hat{j}+4 \hat{k}) \cdot(-2 \hat{i}+\hat{k})`

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