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
\( \lt x \; , \; y \; , \; z \gt \; = \; \lt x_C \; , \; y_C \; , \; z_C \gt + s \lt x_D - x_C \; , \; y_D - y_C \; , \; z_D - z_C \gt \qquad (II)\)

The point of intersection of the lines \( A B \) and \( CD \) is found by solving 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