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Line Of Intersection Of Two Planes Calculator With Slope

Plane Intersection Formula:

\[ \text{Given planes: } a_1x + b_1y + c_1z = d_1 \text{ and } a_2x + b_2y + c_2z = d_2 \] \[ \text{Direction vector: } \vec{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} \] \[ \text{Slope: } m = \frac{v_y}{v_x} \text{ (when projected onto xy-plane)} \]

Plane 1 Equation: a₁x + b₁y + c₁z = d₁

Plane 2 Equation: a₂x + b₂y + c₂z = d₂

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1. What is the Line of Intersection of Two Planes?

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.

2. How Does the Calculator Work?

The calculator uses the following mathematical principles:

\[ \text{Direction vector } \vec{v} = \vec{n_1} \times \vec{n_2} \] \[ \text{Slope } m = \frac{v_y}{v_x} \text{ (in xy-plane projection)} \]

Where:

3. Importance of Plane Intersection

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.

4. Using the Calculator

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.

5. Frequently Asked Questions (FAQ)

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.

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