Hi I need help with this math problem:

Question

The plane that contains the points (8,3,1), (2,6,3), and (4,6,2) has an equation of the form
ax + by + cz = d. Find coefficients for this equation, trying two different approaches to the problem. One method uses vectors, another does not.

I first tried solving the systems of equations and the dot product but I got stuck in the process.

Thank you.

Answer 1

I am not certain what your question is. Have you done
A) Gaussian reduction? Your graphing calculator probably can do that directly for you. Here is an applet that can solve it in .2 seconds. (in your case, row4 and column4 are all zero).

B. On using the dot product, I am uncertain how we can assist you.

Answer 2

I will do the vector method
1. First find two direction vectors in the plane using two different pairs of points
u = (8-2,3-6,1-3) = (6,-3,-2)
v = (8-4,3-6,1-2) = (4,-3,-1)
2. form a normal to these two vectors, (the cross product
I assume you know how to do this, I got (3,2,6)
So the equation of the place is
3x + 2y + 6z = d
plug in one of the points, say (2,6,3)
6 + 12 + 18 = d
d = 36

equation:
3x + 2y + 6z = 36

I will leave it up to you to verify that the other two points also satisfy my equation, they do.

Answer 3

opps, the link for the gaussian reduction: http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/Gauss.html