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I have three points in a plane, i need to find where it intercepts in the x axis , how do I find the coordinate that does that? I have to figure out the plane equation first?

Magorda
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    Plane equation first, then let $y=z=0$. – Vishu Apr 22 '21 at 11:41
  • The points on the plane are of the form $\lambda(Q-P)+\mu(R-P)+P$. Set the second and third coordinate to $0$ and solve for $\lambda,\mu$. If there is a uniqe solution, there is a unique solution, which is $\lambda(Q-P)+\mu(R-P)+P$. If there is not a unique solution, then the plane contains the x-axis. – Levent Apr 22 '21 at 11:46
  • Thanks tavish and levent! – Magorda Apr 22 '21 at 12:00

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No, you don't have to figure out the plane equation.

Let $X(x_0,0,0)$ be the intersection point:

We have:

$$\begin{vmatrix}x_0&x_P&x_Q&x_R\\ 0&y_P&y_Q&y_R\\ 0&z_P&z_Q&z_R\\ 1&1&1&1 \end{vmatrix}=0$$

Why that ? Because $\tfrac16$ of this determinant is equal to the volume of tetrahedron $PQRX$ (see here) and we want this volume to be $0$.

It remains to expand this determinant with respect to the first column and solve for $x_0$:

$$x_0=\dfrac{\begin{vmatrix}x_P&x_Q&x_R\\ y_P&y_Q&y_R\\ z_P&z_Q&z_R\\ \end{vmatrix}}{\begin{vmatrix} y_P&y_Q&y_R\\ z_P&z_Q&z_R\\ 1&1&1 \end{vmatrix}}$$

with the condition that the denominator is not zero, of course.

Jean Marie
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