Answer:
unit normal vector n will be n=(a,b,c) = (4/√171,11/√171,6/√171)
Explanation:
There are several ways to solve this problem
1) build 2 vectors AB and BC such that the vectorial product ABxBC is the orthogonal vector to the plane , then find unit vector
2) since the 3 points belongs to the plane solve a linear system of 4 equation with 4 variables
for the second solution , the equation of the plane with normal vector n=(a,b,c) and containing the point (x₀,y₀,z₀) is
a*(x-x₀)+b*(y-y₀)+c*(z-z₀) =0
and
a²+b²+c² = 1 (unit vector)
then choosing A=(x₀,y₀,z₀)=(1,0,0)
a*(x-1)+b*(y-0)+c*(z-0) =0
for B
a*(3-1)+b*(-1-0)+c*(-3-0) =0
1) 2*a - b - 3*c =0
for C
a*(1-1)+b*(3-0)+c*(-2-0) =0
2) 3*b - 2*c=0 → b= 2/3*c
replacing in 1)
2*a - 2/3*c - 3*c =0
2*a-11/3*c=0 → a=11/6*c
thus
a²+b²+c² = 1
(11/6*c)²+(2/3*c)²+c² = 1
(121/36+4/9+1)*c² = 1
171/36*c²=1 → c= 6/√171
therefore
a=11/6*c = 11/6*6/√171= 11/√171
b=2/3*c= 2/3*6/√171= 4/√171
then the unit normal vector n will be
n=(a,b,c) = (4/√171,11/√171,6/√171)