# Intersection of a Line and a Plane in 3 D Calculator

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