Please help.
Find the vector equation of the line of intersection of the 3 planes represented by this system of equations.
2x - 7y + 5z = 1
6x + 3y - z = -1
-14x - 23y + 13z = 5
Thank you very much!
Solve the system for x and y dependent from z. I've got:
$\displaystyle x=-\frac1{12} - \frac16 z$ and $\displaystyle y = -\frac16 + \frac23 z$
Set z = t. Then you have the parametric equation of the intersection line:
$\displaystyle \left|\begin{array}{l}x = -\frac1{12} - \frac16 t \\ y= -\frac16 + \frac23 t \\z = t \end{array} \right.$
That means the vector equation of the line is:
$\displaystyle (x, y, z) = \left(-\frac1{12}~,~-\frac16~,~0\right) + t \cdot \left(-\frac16~,~ \frac23~,~1\right)$
The first vector is the stationary vector of a point which is located in all three planes and the second vector is the direction vector of the line.