An online calculator to find the point of intersection of a line and a plane in 3D is presented.

Point of Intersection of a LIne and a Plane 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 symmetric form of the above vector equation is given by
\( \dfrac{x - x_A}{ x_B - x_A} = \dfrac{y - y_A}{ y_B - y_A} = \dfrac{z - z_A}{ z_B - z_A} \)

The equation of the 3D plane \( P \) is of the form
\( a x + b y + c z = d \)
A point with coordinates \( x_0 , y_0 , z_0 \) is a point of intersection of the line through \( A B \) and the plane \( P \) if it satisfies two independent equations from (I) and the plane equation. Hence the 3 by 3 systems of equations to solve.
\( \dfrac{x_0 - x_A}{ x_B - x_A} = \dfrac{y_0 - y_A}{ y_B - y_A} \quad (1) \)
\( \dfrac{y_0 - y_A}{ y_B - y_A} = \dfrac{z_0 - z_A}{ z_B - z_A} \quad (2) \)
\( a x_0 + b y_0 + c z_0 = d \quad (3) \)

Use of the Calculator

Enter the coordinates of points \( A \) and \( B \) and the coefficients \( [a,b,c,d] \) included in the equation of the plane 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