pg. 717, #27 (moderate)

Find the equation of the plane passing through (0,0,0), (1,1,1) and (1,2,3).

Solution:

Let v be the vector from (0,0,0) to (1,1,1): v = <1,1,1>. Let w be the vector from (0,0,0) to (1,2,3);

w = <1,2,3>. The normal n to the plane is the cross product of v and w. Then

Since the plane goes through (0,0,0) the normal from for the equation of the plane is

As a check, you should verify that (1,1,1) and (1,2,3) satisfy this equation. They do: