1. What is the Angle Between Planes?

The angle between two planes is defined as the angle between their normal vectors. It's the smallest angle at which one plane would have to be rotated to make it parallel to the other plane.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

• \( \vec{n_1} \) — Normal vector of the first plane
• \( \vec{n_2} \) — Normal vector of the second plane
• \( \cdot \) — Dot product operation
• \( |\vec{v}| \) — Magnitude of vector v

Explanation: The calculator first computes the dot product of the two normal vectors, then divides by the product of their magnitudes to get the cosine of the angle between them.

3. Importance of Angle Calculation

Details: Calculating the angle between planes is important in 3D geometry, computer graphics, architectural design, and engineering applications where spatial relationships between surfaces need to be determined.

4. Using the Calculator

Tips: Enter the x, y, z components of each plane's normal vector. The calculator will compute the smallest angle between the two planes (always between 0° and 90°).

5. Frequently Asked Questions (FAQ)

Q1: What is a normal vector?
A: A normal vector is a vector that is perpendicular to a plane. It defines the plane's orientation in 3D space.

Q2: Can the angle between planes be greater than 90°?
A: The calculator shows the smallest angle between planes (acute angle). The actual angle between normals can be up to 180°, but the angle between planes is always ≤90°.

Q3: What does a 0° angle mean?
A: A 0° angle means the planes are parallel (their normal vectors point in the same or exact opposite directions).

Q4: What does a 90° angle mean?
A: A 90° angle means the planes are perpendicular to each other (their normal vectors are perpendicular).

Q5: Does the order of vectors matter?
A: No, the angle calculation is commutative - the order of the normal vectors doesn't affect the result.