If the points `(1," "1," "p)" "a n d" "(" "3," "0," "1)` be equidistant from the plane ` -> rdot(3 hat i+4 hat j-12 hat k)+13=0` , then find the value of p.

Explanation

Distance between the point and the plane is given by,
`l_(1)=|{:(veca*vecn-d)/(|n|):}|` ...(i)
`veca=hat("i")+hat(j)+hat(k)`,
`vecn=3hat("i")+4hat(j)-12hat(k)`
`veca*vec(n)=3+4-12=-5`
`vecd=-13`
From (i) , `=|(-5+13)/(sqrt(9+16+144))|=(8)/(13)`
Again, `veca=-3hat("i")+0hat(j)+hat(k)`
Them , `l_(2)=|((3hat("i")+0hat(j)+hat(k))*(3hat("i")+4hat(j)-12hat(k))+13)/(sqrt(9+16+144))|`
`=|(-9-12+13)/(13)|`
`=(8)/(13)`
`thereforel_(1)=l_(2)`