1. What are Vector Operations?

Vector operations are mathematical operations performed on vectors, which are quantities that have both magnitude and direction. Common operations include addition, subtraction, and dot product.

2. How Does the Calculator Work?

The calculator performs the following vector operations:

Where:

• \( \vec{a} \) — First vector with components (a_x, a_y, a_z)
• \( \vec{b} \) — Second vector with components (b_x, b_y, b_z)

Explanation: Vector addition combines vectors tip-to-tail, subtraction finds the difference between vectors, and dot product calculates a scalar value representing their alignment.

3. Applications of Vector Operations

Details: Vector operations are fundamental in physics, engineering, computer graphics, and machine learning. They're used in force calculations, 3D modeling, and data analysis.

4. Using the Calculator

Tips: Enter the x, y, z components of both vectors, select the operation type, and click Calculate. Results are shown in component form for addition/subtraction, or as a scalar for dot product.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar value, while cross product gives a vector perpendicular to both input vectors.

Q2: Can I perform operations on 2D vectors?
A: Yes, just set the z-component to 0 for both vectors.

Q3: What does a dot product of 0 mean?
A: It means the vectors are perpendicular (90° angle between them).

Q4: How is vector addition different from scalar addition?
A: Vector addition combines both magnitude and direction, while scalar addition just sums numbers.

Q5: What are unit vectors?
A: Vectors with magnitude 1, often used as basis vectors (i, j, k in 3D space).