Class 12

Math

3D Geometry

Three Dimensional Geometry

Show that the line $r=(2i^+5j^ +7k^)+λ(i^+3j^ +4k^)$ is parallel to the plane $r⋅(i^+j^ −k^)=7$. Also, find the distance between them.

We know that a line $r=a+λb$ is parallel to the plane $r⋅n=q$ only when

this line is perpendicular to the normal to the plane.

So, we must have $b⋅n=0$

Here, $(b⋅n)=(i^+3j^ +4k^)⋅(i^+j^ −k^)=(1+3−4)=0$

Hence, the given line is parallel to the given plane.

Required distance between the line and the plane

$=∣n∣∣a⋅n−q∣ =∣i^+j^ −k^∣∣(2i^+5j^ +7k^)⋅(i^+j^ −k^)−7∣ $

$=1_{2}+1_{2}+(−1)_{2} ∣2+5−7−7∣ $

$=3 ∣−7∣ =3 7 $ units.

$=3 ∣−7∣ =3 7 $ units.