Problem:
Determine if the points P(3, -2, -7), Q(0 , 4, 2), R(-1, 3, -1) and S(5, -1, -3) are coplanar.
Answer in text: Yes
Additional Comments:
Please help me, I have a test tomorrow that basically decides my future.
One very fast and easy way to check if a set of four points are coplanar is by putting them in a matrix and checking the determinant.
Four points, $\displaystyle \left(x_1, y_1, z_1\right),\;\left(x_2, y_2, z_2\right),\;\left(x_3, y_3, z_3\right),\text{ and }\left(x_4, y_4, z_4\right)$ are coplanar if and only if
$\displaystyle \left|\begin{matrix}
x_1 & y_1 & z_1 & 1\\
x_2 & y_2 & z_2 & 1\\
x_3 & y_3 & z_3 & 1\\
x_4 & y_4 & z_4 & 1\\
\end{matrix}\right| = 0$
I don't have a proof of this off-hand, but it should work.
Jhevon is going to kill me for that
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By the way ^^
I suppose you're in high school and because you talked about linear systems, we'll do it this way :
The equation of a plane is $\displaystyle ax+by+cz+d=0$
So write the linear system of equations assuming that P, Q, R and S are in this plane. Then solve for a,b,c and d.
If it has a (unique) solution, it will mean that the points are coplanar I'll let you do the working, because I really have to go to bed
Actually, this is equivalent to Reckoner's method, but his is for higher grades