Vector Calculator

Vectors represent both magnitude and direction. Use this tool to analyze them as follows:

Step 1: Choose the dimension (2D or 3D).

Step 2: Enter the components for Vector A (x, y, z) and Vector B.

Step 3: Select the operation: Addition, Subtraction, Dot Product, or Cross Product.

Step 4: Click "Calculate."

Step 5: Review the detailed breakdown. The calculator will provide: - The resulting vector or scalar value. - The Magnitude (length) of each vector. - The Angle between the two vectors. - The Unit Vector representation for each input.

Vector operations follow specific geometric and algebraic rules:

1. Dot Product (A · B): a₁b₁ + a₂b₂ + a₃b₃. The result is a scalar that represents how much one vector "projects" onto another. 2. Cross Product (A × B): Results in a new vector perpendicular to both A and B, calculated using the determinant of a 3x3 matrix. 3. Magnitude (|A|): √(a₁² + a₂² + a₃²). This is the "length" of the vector. 4. Angle (θ): cos(θ) = (A · B) / (|A||B|).

Example (A=[1,2,3], B=[4,5,6]): Dot Product = (1*4) + (2*5) + (3*6) = 4 + 10 + 18 = 32.

The Vector Calculator is a foundational tool for physics, engineering, and computer graphics. In physics, vectors represent forces, velocities, and accelerations. In game development, they are used to determine movement, lighting, and collisions in 3D environments. This tool simplifies the multi-step calculations required for products and angles, ensuring high precision for navigation and structural analysis. By providing both the scalar (dot) and vector (cross) results, it serves as a comprehensive resource for both students and professional designers.