Plane Intersection Formula:
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When two planes intersect in 3D space, their intersection is typically a straight line. This line can be defined by a direction vector (the cross product of the planes' normal vectors) and a point that lies on both planes.
The calculator uses the following mathematical principles:
Where:
Applications: Finding the intersection line between planes is crucial in 3D geometry, computer graphics, engineering design, and physics simulations. The slope helps understand the line's angle in 2D projections.
Instructions: Enter the coefficients for both plane equations. The calculator will show the direction vector of the intersection line, its slope when projected onto the xy-plane, and a point that lies on both planes.
Q1: What if the planes are parallel?
A: The calculator will show a zero direction vector (⟨0,0,0⟩), indicating the planes are parallel and don't intersect (or are coincident).
Q2: Why is the slope sometimes undefined?
A: When the direction vector has no x-component (vx=0), the line is vertical in the xy-plane projection, making the slope undefined.
Q3: How is the point on the line determined?
A: The calculator attempts to find where the line intersects the xy-plane (z=0). If this isn't possible, it will indicate so.
Q4: Can I use this for planes in different forms?
A: This calculator uses the standard form (ax+by+cz=d). Convert other forms (like point-normal) to standard form first.
Q5: What does the direction vector represent?
A: It shows the line's direction in 3D space. The slope is just its 2D projection for visualization purposes.