Line of Intersection of Two Planes Calculator
Solve line intersection two planes problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
Direction = n\u2081 \u00D7 n\u2082 (cross product of normal vectors)
The direction vector of the intersection line is the cross product of the two plane normals. A point on the line is found by setting one variable to zero and solving the remaining 2x2 system. The parametric form is P = P\u2080 + t * d.
Worked Examples
Example 1: Perpendicular Planes
Problem: Find the intersection of planes x + y + z = 6 and x - y + z = 2.
Solution: Normal vectors: n1 = (1,1,1), n2 = (1,-1,1)\nDirection = n1 x n2 = (1*1-1*(-1), 1*1-1*1, 1*(-1)-1*1) = (2, 0, -2)\nSet z=0: x+y=6 and x-y=2 => x=4, y=2\nPoint: (4, 2, 0)\nParametric: x=4+2t, y=2, z=-2t\nAngle: cos(theta) = |1-1+1|/(sqrt(3)*sqrt(3)) = 1/3, theta = 70.53\u00B0
Result: Line: (4+2t, 2, -2t) | Direction: (2, 0, -2) | Angle: 70.53\u00B0
Example 2: General Plane Intersection
Problem: Find the intersection of 2x + y - z = 3 and x - y + 2z = 1.
Solution: Normal vectors: n1 = (2,1,-1), n2 = (1,-1,2)\nDirection = n1 x n2 = (1*2-(-1)*(-1), (-1)*1-2*2, 2*(-1)-1*1) = (1, -5, -3)\nSet z=0: 2x+y=3 and x-y=1 => 3x=4, x=4/3, y=1/3\nPoint: (1.333, 0.333, 0)\nAngle: cos(theta) = |2-1-2|/(sqrt(6)*sqrt(6)) = 1/6, theta = 80.41\u00B0
Result: Line: (1.333+t, 0.333-5t, -3t) | Direction: (1, -5, -3) | Angle: 80.41\u00B0
Frequently Asked Questions
How do you find the line of intersection of two planes?
To find the line where two planes intersect, you need two things: a direction vector for the line and a point on it. The direction vector is the cross product of the two planes' normal vectors: d = n1 x n2. This works because the intersection line must be perpendicular to both normal vectors simultaneously. To find a specific point on the line, set one variable (say z) to zero and solve the resulting system of two equations in two unknowns. The parametric equation of the line is then P = P0 + t*d, where P0 is the point found and t is a parameter. If the cross product is the zero vector, the planes are parallel and do not intersect in a line.
What does it mean when two planes are parallel?
Two planes are parallel when their normal vectors are scalar multiples of each other, meaning n1 = k * n2 for some scalar k. Geometrically, parallel planes never intersect and maintain a constant distance between them everywhere. The distance between parallel planes a1*x + b1*y + c1*z = d1 and a1*x + b1*y + c1*z = d2 is |d1 - d2| / sqrt(a1\u00B2 + b1\u00B2 + c1\u00B2). If the planes are parallel AND have the same constant term (after normalization), they are coincident, meaning they are the same plane and every point on one is on the other. Line of Intersection of Two Planes Calculator detects both cases by checking whether the cross product of the normal vectors is the zero vector.
How do you write the parametric equations of the intersection line?
The parametric equations express each coordinate as a function of a single parameter t. Given a point P0 = (x0, y0, z0) on the line and the direction vector d = (dx, dy, dz), the parametric form is: x = x0 + dx*t, y = y0 + dy*t, z = z0 + dz*t, where t ranges from negative infinity to positive infinity. Each value of t gives a unique point on the line. At t = 0, you get the reference point P0. Positive t values give points in one direction, and negative t values give points in the opposite direction. The parametric form is the most general and useful representation of a line in 3D space because it handles all orientations, including lines parallel to coordinate planes.
What is the symmetric form of a line equation in 3D?
The symmetric form eliminates the parameter t from parametric equations by setting them equal: (x - x0)/dx = (y - y0)/dy = (z - z0)/dz. This form is compact but has limitations: it cannot be used when any component of the direction vector is zero, because that would create division by zero. In such cases, the zero component means the corresponding coordinate is constant. For instance, if dz = 0, the line lies in the plane z = z0, and you would write (x - x0)/dx = (y - y0)/dy with z = z0 separately. The symmetric form is commonly used in textbooks for compactness but the parametric form is more practical for computations.
How do you calculate the angle between two planes?
The angle between two planes equals the angle between their normal vectors. Using the dot product formula: cos(theta) = |n1 * n2| / (|n1| * |n2|), where the absolute value ensures the acute angle is returned (since normals can point in opposite directions for the same plane). If the dot product is zero, the planes are perpendicular (90 degrees apart). If the absolute value of the cosine is 1, the planes are parallel (0 degrees apart). The angle between planes is also called the dihedral angle and has applications in crystallography, architecture, and origami design. Note that two intersecting planes always form two supplementary dihedral angles that sum to 180 degrees.
What are practical applications of plane intersection?
Finding the intersection of planes has numerous practical applications in science and engineering. In computer graphics, clipping algorithms determine where objects cross viewing planes. In structural engineering, the intersection of roof planes determines ridge lines and valley lines in roof design. In geology, the intersection of fault planes and bedding planes determines outcrop patterns. In crystallography, crystal faces are defined as planes, and their intersections form crystal edges. GPS and surveying use plane intersections for triangulation in three dimensions. In manufacturing, CNC machines compute tool paths by intersecting cutting planes with workpiece surfaces. Even in aviation, the intersection of navigational planes determines flight path intersections.