The perpendicular distance between the line `vecr = 2hati-2hatj+3hatk+lambda(hati-hatj+4hatk)` and the plane `vecr.(hati + 5hatj + hatk) = 5` is :

Options

A. `10/9`
B. `10/(3sqrt(3))`
C. `10/3`
D. none of these

Answer

B

Explanation

Clearly, the given line passes through the point `veca=2hati-2hatj+3hatk` and is parallel to the vector `vecb=hati-hatj+4hatk.`
The plane is normal to the vector `vecn=hati+5hatj+hatk`.
We have `vecb.vecn=1-5+4=0`
So, the line is parallel to the plane.
`:.` Required distance
`=` Length of the perpendicular from a point on the line to the given plane.
`=` Length of the perpendicular from `(2hati-2hatj+3hatk)` to the given plane.
`|((2hati-2hatj+3hatk).(hati+5hatj+hatk)-5)/(sqrt(1+25+1))|=|(2-10+3-5)/(3sqrt(3))|=10/(3sqrt(3))`