# Equations of the line of intersection of two planes

This online calculator finds the equations of a straight line given by the intersection of two planes in space. The calculator displays the canonical and parametric equations of the line, as well as the coordinates of the point belonging to the line and the direction vector of the line.

This online calculator is designed to find solutions to problems that can be formulated as follows:

Find the parametric equations for the line of intersection of the planes.
$\left\{ A_1x+B_1y+C_1z+D_1=0 \atop A_2x+B_2y+C_2z+D_2=0$.

You enter the coefficients of the general equations of the planes, A₁, B₁, C₁, D₁ and A₂, B₂, C₂, D₂, in the form below, the calculator displays the equations of the line of intersection in parametric and canonical forms, as well as the found point belonging to the line and the direction vector of the line.

Pay attention, if the problem gives the equations of the planes in the form
$\left\{ A_1x+B_1y+C_1z=D_1 \atop A_2x+B_2y+C_2z=D_2$,
you need to change the sign when entering the coefficients D₁ and D₂.

A bit of theory, as usual, can be found below the calculator

#### Equations of the line of intersection of two planes

Point lying on the line

Direction vector of the line.

Canonical equation of the line

Parametric equations of the line

Digits after the decimal point: 2

### Canonical equation of a straight line given by the intersection of two planes

If the planes intersect, then the system of planes equations given at the beginning of the article defines a straight line in space. To write the equations of this line in the canonical form, you need to find a point on the line of intersection and a direction vector.

Point belonging to the line also belongs to each of the planes, that is, it is one of the solutions to the system of equations above. To find a point belonging to a straight line, we switch from a system of two equations with three unknowns to a system of two equations with two unknowns, arbitrarily taking any coordinate of the point as zero. As a rule, when solving problems, we choose the coordinate, which, when zeroed, results in the integer solution of a remaining system of two equations (a result with integer coefficients looks much prettier). The calculator takes this fact into account and also tries to find an integer solution, zeroing out all the coordinates one by one.

The direction vector of the straight line is orthogonal to the normal vectors of the planes, which are given by the coefficients A, B and C in the general equation of a plane $Ax+By+Cz+D=0$. Thus, it can be found as the result of the cross product of the normal vectors of the planes: $\hat{p}=\hat{n_1}\times\hat{n_2}$.

The point $(x_0;y_0;z_0)$ and the vector $(p_1;p_2;p_3)$ give us the canonical equation of the line in space:

$\frac{x-x_0}{p_1}=\frac{y-y_0}{p_2}=\frac{z-z_0}{p_3}$

There are special cases when one or two coordinates of the direction vector are equal to zero.

If two coordinates are equal to zero, the direction vector is collinear to one of the coordinate axes. Accordingly, the points of the straight line can take any value along this axis, while the values ​​along the other two axes will be constant. For example, if the two zero coordinates are y and z, the canonical equations of a straight line will look like this:
$y-y_0=0; z-z_0=0$

If only one coordinate is equal to zero, the direction vector lies in one of the coordinate planes (planes formed by pairs of coordinate axes), the value of the coordinate along the third axis orthogonal to this plane (just the one for which the coordinate of the direction vector is zero) will again be constant. For example, if the zero coordinate is x, then the canonical equations of the line will look like this:
$x-x_0=0; \frac{y-y_0}{p_2}=\frac{z-z_0}{p_3}$

These cases are also taken into account by the calculator.

### Parametric equations of a straight line given by the intersection of two planes

Knowing the point lying on the line and the direction vector of the line, it is easy to write down the parametric equations of the line. For a point $(x_0;y_0;z_0)$ belonging to the line and a direction vector $(p_1;p_2;p_3)$, the parametric equations of the line look like this:
$x=p_1t+x_0\\y=p_2t+y_0\\z=p_3z+z_0$

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PLANETCALC, Equations of the line of intersection of two planes