# Intersection of Two Lines in 3 D Calculator



An online calculator to find the point of intersection of two lines in 3D is presented.

## Point of Intersection of Two Lines in 3D

The equation in vector form of a line throught the points $A(x_A \; , \; y_A \; , \; z_A)$ and $B(x_B \; , \; y_B \;, \; z_B)$ is written as
$\lt x \; , \; y \; , \; z \gt \; = \; \lt x_A \; , \; y_A \; , \; z_A \gt + t \lt x_B - x_A \; , \; y_B - y_A \; , \; z_B - z_A \gt \qquad (I)$
The equation in vector form of a line throught the points $C(x_C \; , \; y_C \; , \; z_C)$ and $D(x_D \; , \; y_D \;, \; z_D)$ is written as
$\lt x \; , \; y \; , \; z \gt \; = \; \lt x_C \; , \; y_C \; , \; z_C \gt + s \lt x_D - x_C \; , \; y_D - y_C \; , \; z_D - z_C \gt \qquad (II)$

A point with coordinates $x_0 , y_0 , z_0$ is a point of intersection of the lines $A B$ and $CD$ if it satisfies the equation:
$\lt x_A \; , \; y_A \; , \; z_A \gt + t \lt x_B - x_A \; , \; y_B - y_A \; , \; z_B - z_A \gt \; = \; \lt x_C \; , \; y_C \; , \; z_C \gt + s \lt x_D - x_C \; , \; y_D - y_C \; , \; z_D - z_C \gt$
which gives the following
systems of 3 equations and 2 unknowns $t$ and $s$.
$x_A + t \; ( x_B - x_A ) = x_C + s \; (x_D - x_C )$
$y_A + t \; ( y_B - y_A ) = y_C + s \; (y_D - y_C )$
$z_A + t \; ( z_B - z_A ) = z_C + s \; (z_D - z_C )$
We solve two equations for $t$ and $s$ and verify whether the third one is satisfied or not.

## Use of the Calculator

Enter the coordinates of points $A$, $B$, $C$ and $D$ as real numbers separated by commas as shown below then press "Calculate".

$A(x_A \; , \; y_A \; , \; z_A)$: (
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$B(x_B \; , \; y_B \; , \; z_B)$: (
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$C(x_C \; , \; y_C \; , \; z_C)$: (
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$B(x_D \; , \; y_D \; , \; z_D)$: (
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