How do you find a unit vector perpendicular to a 3-D plane formed by points (1,01),(0,2,2) and (3,3,0)?
Our strategy will be to find two vectors in the plane, take their cross product to find a vector perpendicular to both of them (and thus to the plane), and then divide that vector by its measure to make it a unit vector.
Step 1) Find two vectors in the plane.
We will do this by finding the vector from
Step 2) Find a vector perpendicular to the plane.
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 a vector perpendicular to both, we will use the cross product of
#= |(hat(i), hat(j), hat(k)), (-1, 2, 1), (2, 3, -1)|#
#=(-5, 1, -7)#
Step 3) Turn
A unit vector is a vector whose measure is
As multiplying by a scalar does not change the direction of a vector, this will be a unit vector perpendicular to the plane. Proceeding,
Thus, our final result is