Find two vectors in the plane. We will do this by finding the vector from (1,−1,0)(2,1,−2) and from (1,−1,0)(−1,1,2). As all three points are in the plane, so will each of those vectors.
→
u1
=(1,−1,0)−(2,1,−2)=(−1,−2,2)
→
u2
=(1,−1,0)−(−1,1,2)=(2,−2,2) If a vector is perpendicular to two vectors in a plane, it must be perpendicular to the plane itself. As the cross product of two vectors produces vector perpendicular to both, we will use the cross product of
→
u1
and
→
u2
to find a vector
→
u
perpendicular to the plane containing them. ⇒
→
u
=
→
u1
×
→
u2
⇒
→
u
=|
→
i
→
j
→
k
−1
−2
2
2
−2
2
| ⇒
→
u
=
→
i(
−4+4)−
→
j(
−2−4)+
→
k
(2+4) ⇒
→
u
=6
→
j
+6
→
k
Hence, the vector perpendicular to the plane passing through (1,−1,0)(2,1,−2) and (−1,1,2) is