Find an equation of the plane that passes through the points $(1, 2, 5), (5, 4, 8) \textrm{ and } (2, 4, 8)$

Question

Find an equation of the plane that passes through the points $(1, 2, 5), (5, 4, 8) $, and $(2, 4, 8)$.

$v_1 = [1,2,5] - [5,4, 8] = [-4, -2, -3]$

$v_2 = [2,4,8] - [5,4,8] = [-3,0,0]$

$v_1 \times v_2 = [0,9,-6]$

thus,

$ax + by + cz = d, a = 0, b = 9, c = -6$

to get d we plug $0(5) + 9(4) - 6(8) = -12$

Therefore the equation is $9y - 6z = -12$

Right?

Answer 1

Yes, your answer is correct.

Note that if $P(x,y,z)$ is on the plane passing through your given points then then the volume generated by vectors $$ (x-1,y- 2,z- 5), (x-5,y- 4,z- 8), (x-2,y- 4,z- 8)$$ is zero.

Thus the equation of the plain is:

$$ \det \begin{bmatrix} x-1&y-2 &z-5\\x-5&y-4&z-8\\x-2&y-4&z-8\end{bmatrix} =0$$

Answer 2

Another useful formula is

$$ \begin{vmatrix} 1&1&1&1\\ 1&2&5&x\\ 5&4&8&y\\ 2&4&8&z \end{vmatrix}=0 $$

We can see that in the 1st row there are 1-s, then the coordinates of the points followed by $x,y,z$, respectively. This is, in fact equivalent to the formula given by the previous answer.