Minh Nguyen                              		mtn8286
Week 1                                   		1.2.8

http://calclab.math.tamu.edu/~mtn8286/1_2_8.html


Problem:
Find an equation and a parametric representation for the plane
passing through the points (1,0,1), (2,3,1), (5,4,5).


Solution:
(Note: Vectors are denoted in bold italics as well as arrow notation)

The parametric form of a plane is:
  
	x=su1+tu2+x0     where x0 is a point in the plane, and u1 
			and u2 are vectors tangent to the plane.

Since the difference between any two points on the plane is a vector 
parallel to the plane, we can find u1 and u2. We can arbitrarily 
subtract the first given vector from the other two to give us:

	u1 = (1,3,0),	u2 = (4,4,4)

By choosing the first vector as x0, the parametric representation is:

	

The equation of a plane is of the form:


	ax+by+cz=d      where d=n.x0 for any point x0 in the plane,
			and n is perpendicular to the plane.

Therefore, n can be obtained by u1xu2(cross product):
	

By this method,the equation of the plane is:  12x-4y-8z=4  

And simplified further, the equation becomes:  3x-y-2z=1