Show that the plane whose vector equation is ` vec rdot( hat i+2 hat j- hat k)=1.` and the line whose vector equation is ` vec r=(- hat i+ hat j+ hat k)+lambda(2 hat i+ hat j+4 hat k)` are parallel. Also, find the distance between them.

Explanation

A line contains line if dot product of normal vector of plane and direction
cosine of line is 0 and point on line satisfies the plane.

We have:

` vec rdot( hat i+2 hat j- hat k)=1.` ....(i)

Normal vector `=n
=hat i+2 hat j- hat k`

Equation of line is

`r
=-i^+j^+k^​+λ(2i^+j^​+4k^)`

...