1. What is the Angle Between Two Vectors?

The angle between two vectors is the smallest angle between them when their tails coincide. It's calculated using the dot product formula and provides important geometric information about their relationship.

2. How Does the Calculator Work?

The calculator uses the vector angle formula:

Where:

• \( \vec{v}_1 \cdot \vec{v}_2 \) — Dot product of vectors
• \( \|\vec{v}_1\| \) — Magnitude of vector 1
• \( \|\vec{v}_2\| \) — Magnitude of vector 2
• \( \arccos \) — Inverse cosine function

Explanation: The formula comes from the geometric interpretation of the dot product, which relates the angle between vectors to their magnitudes and dot product.

3. Importance of Vector Angles

Details: The angle between vectors indicates their relative direction. A 0° angle means they point in the same direction, 90° means they're perpendicular, and 180° means they point in opposite directions.

4. Using the Calculator

Tips: Enter the x, y, z components for both vectors. The calculator works for both 2D (set z=0) and 3D vectors. The result is given in degrees (0° to 180°).

5. Frequently Asked Questions (FAQ)

Q1: What does a 90° angle indicate?
A: A 90° angle means the vectors are orthogonal (perpendicular) to each other.

Q2: Can the angle be greater than 180°?
A: No, the angle between vectors is always the smallest angle between them (0° to 180°).

Q3: What if one vector has zero magnitude?
A: The angle is undefined for zero vectors as direction cannot be determined.

Q4: How is this different from 2D vectors?
A: The same formula works for 2D vectors - just set the z-component to zero.

Q5: What practical applications does this have?
A: Used in physics, computer graphics, robotics, and anywhere vector relationships are important.