Line Of Two Planes Calculator

Line of Intersection of Two Planes:

\[ \text{Given planes: } a_1x + b_1y + c_1z = d_1 \text{ and } a_2x + b_2y + c_2z = d_2 \] \[ \text{Direction vector: } \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} \]

Plane 1 (a):

Plane 1 (b):

Plane 1 (c):

Plane 1 (d):

Plane 2 (a):

Plane 2 (b):

Plane 2 (c):

Plane 2 (d):

Direction Vector:

Point on Line:

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1. What is the Line of Two Planes?

The line of intersection of two planes is the set of all points that lie on both planes. In 3D space, two non-parallel planes will always intersect in a line.

2. How Does the Calculator Work?

The calculator finds the line of intersection by:

\[ \text{Direction vector: } \vec{v} = \vec{n_1} \times \vec{n_2} \] \[ \text{Point on line: Solution to } a_1x + b_1y + c_1z = d_1 \text{ and } a_2x + b_2y + c_2z = d_2 \]

Where:

\( \vec{n_1} \) — Normal vector of first plane [a₁, b₁, c₁]
\( \vec{n_2} \) — Normal vector of second plane [a₂, b₂, c₂]
\( \times \) — Cross product operation

3. Importance of Line of Intersection

Details: Finding the line of intersection is fundamental in 3D geometry, computer graphics, and engineering applications where spatial relationships between surfaces need to be determined.

4. Using the Calculator

Tips: Enter the coefficients of both plane equations. The calculator will return the direction vector of the line and one point that lies on both planes.

5. Frequently Asked Questions (FAQ)

Q1: What if the planes are parallel?
A: The calculator will show a zero direction vector if the planes are parallel (including coincident planes).

Q2: Why is only one point returned?
A: The line is infinite, so we return one representative point (when z=0) plus the direction vector that defines the line.

Q3: Can I use this for planes in different forms?
A: The calculator currently only accepts the standard form (ax + by + cz = d). Convert other forms to standard form first.

Q4: How accurate are the results?
A: Results are accurate to numerical precision, but parallel planes may show very small non-zero vectors due to floating-point arithmetic.

Q5: What applications use this calculation?
A: Computer graphics, CAD software, robotics path planning, and 3D modeling all rely on plane intersection calculations.