# 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".

$A(x_A \; , \; y_A \; , \; z_A)$: (
)

$B(x_B \; , \; y_B \; , \; z_B)$: (
)

Plane Coefficients: $[ a \; , \; b \; , \; c \; , \; d ]$ = [
]