Line Of Intersection Of Two Planes
Find the parametric equation of the line where two planes intersect in 3D space. Free online line of intersection calculator for linear algebra, geometry, and multivariable calculus.
About This Calculator
The Line of Intersection of Two Planes Calculator computes the parametric equation of the line where two planes intersect in three-dimensional space. This tool is essential for students and professionals working in linear algebra, multivariable calculus, 3D geometry, computer graphics, and engineering applications where plane intersections arise.
Given two planes in the standard form a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2, the intersection is a straight line (unless the planes are parallel). The calculator computes the direction vector of the line using the cross product of the normal vectors: v = n1 x n2 = (b1c2 - c1b2, c1a2 - a1c2, a1b2 - b1a2). It then finds a common point on both planes by solving the system with one coordinate set to zero. The result is displayed as a parametric equation: (x, y, z) = P0 + lambdav.
If the planes are parallel (cross product is zero), no intersection line exists and the calculator returns no result -- this correctly indicates that the planes either are parallel and distinct or coincident.
Frequently Asked Questions
What is the line of intersection of two planes?
The line of intersection of two planes is the set of all points that lie on both planes simultaneously. In 3D space, two non-parallel planes always intersect in a straight line. The line is described by a parametric equation (x, y, z) = P0 + lambdav, where P0 is a common point on both planes and v is the direction vector perpendicular to both normal vectors.
How do you find the line of intersection of two planes?
To find the line of intersection of two planes, first compute the cross product of the normal vectors to get the direction vector of the line. Then find a common point on both planes by setting one coordinate to zero and solving the remaining system of two equations. The parametric equation combines the common point and the direction vector.
When do two planes not intersect?
Two planes do not intersect when they are parallel, meaning their normal vectors are scalar multiples of each other. If parallel planes are distinct, they never meet; if they coincide, every point on one plane is also on the other. The calculator detects parallel planes by checking whether the cross product of the normal vectors is zero.
What is the parametric equation of a line in 3D?
The parametric equation of a line in 3D space is (x, y, z) = (x0, y0, z0) + lambda(v_x, v_y, v_z), where (x0, y0, z0) is a point on the line, (v_x, v_y, v_z) is the direction vector, and lambda is a free parameter that can take any real value. Each value of lambda gives a different point on the line.
Can the intersection of two planes be empty?
Yes, if the two planes are parallel but not coincident, they have no intersection. Parallel planes have normal vectors that are scalar multiples of each other but different constant terms. For example, x + y + z = 1 and x + y + z = 5 are parallel planes with no common points.
What is the direction vector of the intersection line?
The direction vector of the intersection line is the cross product of the two normal vectors: v = n1 x n2. This vector is perpendicular to both normal vectors, so it lies in both planes and traces the line of intersection. For planes a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2, the direction vector is (b1c2 - c1b2, c1a2 - a1c2, a1b2 - b1a2).
How does the calculator handle parallel or coincident planes?
When the cross product of the normal vectors is zero, the calculator detects that the planes are parallel. In this case, no line of intersection exists (or the planes coincide). The calculator will not display a result, indicating the planes do not intersect in a line.
Is the line of intersection unique for two planes?
Yes, two distinct non-parallel planes intersect in exactly one unique line. However, the parametric representation of that line is not unique -- any point on the line can serve as the anchor point P0, and any scalar multiple of the direction vector describes the same line. The calculator chooses one convenient representation by setting one coordinate to zero.