# Thread: Vector(3-Space) Word Problem Help

1. ## Vector(3-Space) Word Problem Help

Problem:

Determine if the points P(3, -2, -7), Q(0 , 4, 2), R(-1, 3, -1) and S(5, -1, -3) are coplanar.

2. Originally Posted by Morphayne
Problem:

Determine if the points P(3, -2, -7), Q(0 , 4, 2), R(-1, 3, -1) and S(5, -1, -3) are coplanar.
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, $\left(x_1, y_1, z_1\right),\;\left(x_2, y_2, z_2\right),\;\left(x_3, y_3, z_3\right),\text{ and }\left(x_4, y_4, z_4\right)$ are coplanar if and only if

$\left|\begin{matrix}
x_1 & y_1 & z_1 & 1\\
x_2 & y_2 & z_2 & 1\\
x_3 & y_3 & z_3 & 1\\
x_4 & y_4 & z_4 & 1\\
\end{matrix}\right| = 0$

I don't have a proof of this off-hand, but it should work.

3. 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 $ax+by+cz+d=0$

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

4. Here is an easy if tedious way to check.
If $\overrightarrow {PS} \cdot \left( {\overrightarrow {PQ} \times \overrightarrow {PR} } \right) = 0$ then the four points are coplanar.