Given - and

To prove – The line and the plane are parallel &

To find – The distance between them

Direction ratios of the line = (1, 3, 4)

Direction ratios of the normal of the plane = (1, 1, - 1)

Formula to be used – If (a, b, c) be the direction ratios of a line and (a’, b’, c’) be the direction ratios of the normal to the plane, then, the angle between the two is given by

The angle between the line and the plane

Hence, the line and the plane are parallel.

Now, the equation of the plane may be written as x + y - z = 7.

Tip – If ax + by + c + d = 0 be a plane and be a line vector, then the distance between them is given by

The distance between the plane and the line

units