**1.** We can use vectors and the cross product to construct planes quite easily.

**2. Determining a Plane From a Point and Normal Vector.** Given a vector

(1)

and a point , there exists a unique plane which is perpindicular to and contains .

How? Well, let be an arbitrary point on the plane. Then the vector would be parallel to the plane. Being parallel to the plane implies its orthogonal to :

(2)

This gives us an equation

(3)

So any point lies on the plane if it satisfies this equation.

**3. Determine a Plane from Three Points.** Given three points , , , find a plane containing these points.

We construct the vectors and . Take the cross product, which produces the normal vector

(4)

If we write and , then

(5)

describes the plane. (It follows from our last construction of the plane.)

**Exercise 1** [Math.SE]**.** Determine the plane containing point and going through the intersection line of the planes and .

**Exercise 2** [Math.SE]**.** Let be the line given by , , . Then intersects the plane at the point . Find parametric equations for the line through which lies on plane and is perpendicular to .

**Exercise 3** [Math.SE]**.** Find normal form equation of plane containing and Line: where is some parameter.