Find the equation of the plane passing through `A(2,2,-1),B(3,4,` `2)a n dC(7,0,6)dot` Also find a unit vector perpendicular to this plane.

Explanation

Here `(x_(1), y_(1), z_(1))-= (2, 2,-1), (x_(2), y_(2), z_(2))-= (3, 4, 2) and (x_(3), y_(3), z_(3))-= (7, 0, 6)`
Then the equation of the plane is
`" "|{:(x-x_(1),,y-y_(1),,z-z_(1)),(x_(2)-x_(1),,y_(2)-y_(1),,z_(2)-z_(1)),(x_(3)-x_(1),,y_(3)-y_(1),,z_(3)-z_(1)):}|=0or |{:(x-2,,y-2,,z-(-1)),(3-2,,4-2,,2-(-1)),(7-2,,0-2,,6-(-1)):}|=0`
or `" "5x+2y-3z=17`
A normal vector to this plane is `vecd=5hati+2hatj-3hatk`
Therefore, a unit vector normal to (i) is given by
`" "hatn=(vecd)/(|vecd|)=(5hati+2hatj-3hatk)/(sqrt(25+4+9))=(1)/(sqrt(38))(5hati+2hatj-3hatk)`