Perpendicular 3D Vector Calculator

Dot Product Formula for Perpendicular Vectors:

\[ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z = 0 \]

Vector A (x):

Vector A (y):

Vector A (z):

Vector B (x):

Vector B (y):

Vector B (z):

Dot Product:

Are Vectors Perpendicular?

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1. What is Perpendicularity in 3D Vectors?

Two vectors are perpendicular (orthogonal) in 3D space when the angle between them is exactly 90 degrees. This occurs when their dot product equals zero, regardless of their magnitudes.

2. How Does the Calculator Work?

The calculator uses the dot product formula:

\[ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z \]

Where:

\( \vec{A} = (A_x, A_y, A_z) \) — First 3D vector
\( \vec{B} = (B_x, B_y, B_z) \) — Second 3D vector
If result = 0 → Vectors are perpendicular
If result ≠ 0 → Vectors are not perpendicular

3. Importance of Dot Product

Details: The dot product is a fundamental operation in vector mathematics that measures the similarity of direction between two vectors. A zero dot product indicates complete orthogonality, which is crucial in physics, computer graphics, and engineering applications.

4. Using the Calculator

Tips: Enter the x, y, and z components for both vectors. The calculator will compute the dot product and determine if the vectors are perpendicular. Use decimal values for precise calculations.

5. Frequently Asked Questions (FAQ)

Q1: What if the dot product is very close to zero but not exactly zero?
A: Due to floating-point precision, very small values (like 1e-10) can be considered effectively zero for practical purposes.

Q2: Can zero vectors be perpendicular?
A: Technically, the zero vector is orthogonal to every vector, but this is a degenerate case usually excluded from practical considerations.

Q3: Does perpendicularity depend on vector magnitude?
A: No, perpendicularity depends only on the angle between vectors, not their lengths.

Q4: How is this different from 2D vector perpendicularity?
A: The concept is the same, but 3D vectors have an additional z-component in the calculation.

Q5: What are some real-world applications?
A: Used in computer graphics (normal vectors), physics (force components), and engineering (structural analysis).