Vector Calculator

Vector Operations Made Simple

This calculator is designed for students and professionals in physics, engineering, and mathematics. It performs common operations on two 3D vectors, including calculating the magnitude (length) of each vector, their dot product, and their cross product.

The Formulas Explained

For two vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃):

• Magnitude (||A||): √(a₁² + a₂² + a₃²) - The length of the vector.
• Dot Product (A · B): a₁b₁ + a₂b₂ + a₃b₃ - A scalar value representing the projection of one vector onto another.
• Cross Product (A × B): (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁) - A new vector that is perpendicular to both A and B.

How to Use the Calculator

1. Enter Vector A: Input the i, j, and k components of the first vector.
2. Enter Vector B: Input the i, j, and k components of the second vector.
3. Calculate: The tool will display the magnitudes of both vectors, their dot product, and the resulting vector from their cross product.

Real-World Example

• Vector A = (2, 3, 4)

• Vector B = (5, 6, 7)

• Magnitude of A: √(2²+3²+4²) = √(4+9+16) = √29 ≈ 5.385

• Dot Product: (2*5) + (3*6) + (4*7) = 10 + 18 + 28 = 56

• Cross Product: (3*7 - 4*6, 4*5 - 2*7, 2*6 - 3*5) = (21-24, 20-14, 12-15) = (-3, 6, -3)

Frequently Asked Questions (FAQ)

• What is the dot product used for? It's used to find the angle between two vectors and to calculate work in physics (Work = Force · Displacement).
• What is the cross product used for? It's used to find a vector that is perpendicular to two other vectors, which is useful in calculating torque and determining orientation in 3D space.
• Does this work for 2D vectors? Yes. For 2D vectors, simply set the 'k' component of both vectors to 0.