# Thread: Line of intersection of 3 planes.

1. ## Line of intersection of 3 planes.

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!

2. Originally Posted by mathstudent43

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.

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### how to find line of intersection of three planes

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