Find the length of perpendicular from the origin to the plane `vecr.(3hati-12hatj-4hatk)+39=0`. Also write the unit normal vector from the origin to the plane.

Answer

`p=3,hatn=(-3/13hati+12/13hatj+4/13hatk)`

Explanation

Given equation in standard form is `vecr.(-3hati+12hatj+4hatk)=39`.
This is `vecr.vecn=p`, where `vecn=(-3hati+12hatj+4hatk)` and p=39.
`|vecn|=sqrt((-3)^(2)+(12)^(2)+4^(2))=sqrt(9+144+16)=sqrt(169)=13`.
`therefore vecr.(-3/13hati+12/13hatj+4/13hatk)=3`.
Hence, p=3 and `vecn=-3/13hati+12/13hatj+4/13hatk`.