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Line-Plane Intersection

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LinePlaneIntersection

The plane determined by the points x_1, x_2, and x_3 and the line passing through the points x_4 and x_5 intersect in a point which can be determined by solving the four simultaneous equations

0=|x y z 1; x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1|
(1)
x=x_4+(x_5-x_4)t
(2)
y=y_4+(y_5-y_4)t
(3)
z=z_4+(z_5-z_4)t
(4)

for x, y, z, and t, giving

t=-(|1 1 1 1; x_1 x_2 x_3 x_4; y_1 y_2 y_3 y_4; z_1 z_2 z_3 z_4|)/(|1 1 1 0; x_1 x_2 x_3 x_5-x_4; y_1 y_2 y_3 y_5-y_4; z_1 z_2 z_3 z_5-z_4|).
(5)

This value can then be plugged back in to (2), (3), and (4) to give the point of intersection (x,y,z).

See also

Line, Line-Line Intersection, Plane, Plane-Plane Intersection

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Cite this as:

Weisstein, Eric W. "Line-Plane Intersection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Line-PlaneIntersection.html

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