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".

More References and Links

Problems on Lines in 3D with Detailed Solutions
Systems of Equation Solvers and Calculators