# 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)?

##### 1 Answer

#### Explanation:

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)|#

#=(2(-1)-1(3))hat(i)-((-1)(-1)-(1)(2))hat(j)+((-1)(3)-(2)(2))hat(k)#

#=-5hat(i)+hat(j)-7hat(k)#

#=(-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