Vector Calculator

About Vector Calculations

Vectors are mathematical objects that have both magnitude (length) and direction. They are fundamental in physics, engineering, computer graphics, and many other fields. This calculator helps you perform various operations with vectors in 3D space.

Vector Operations

The calculator supports the following vector operations:

• Vector Addition: Combines two vectors to produce a third vector (A + B)
• Vector Subtraction: Finds the difference between two vectors (A - B)
• Dot Product: Calculates the scalar product of two vectors (A · B)
• Cross Product: Computes the vector product of two vectors (A × B)
• Vector Magnitude: Determines the length of a vector (||A||)
• Angle Between Vectors: Finds the angle θ between two vectors
• Unit Vector: Calculates a vector with the same direction but length 1 (A/||A||)
• Projection: Projects vector A onto vector B (proj_BA)

Mathematical Formulas

The calculator uses these mathematical formulas for vector operations:

• Addition: A + B = (A_x+B_x, A_y+B_y, A_z+B_z)
• Subtraction: A - B = (A_x-B_x, A_y-B_y, A_z-B_z)
• Dot Product: A · B = A_xB_x + A_yB_y + A_zB_z
• Cross Product: A × B = (A_yB_z-A_zB_y, A_zB_x-A_xB_z, A_xB_y-A_yB_x)
• Magnitude: ||A|| = √(A_x² + A_y² + A_z²)
• Angle: θ = cos⁻¹((A·B)/(||A|| ||B||))
• Unit Vector: Â = A/||A||
• Projection: proj_BA = ((A·B)/(B·B)) B

Applications

Vector operations have numerous applications:

• Physics: Calculating forces, velocities, and accelerations
• Engineering: Structural analysis, fluid dynamics
• Computer Graphics: 3D modeling, lighting calculations
• Navigation: GPS systems, flight paths
• Machine Learning: Feature vectors, similarity measures

Tips for Use

When using this calculator:

• For 2D vectors, set the Z component to 0
• The cross product is only defined for 3D vectors
• The angle between vectors is always between 0° and 180°
• A unit vector has magnitude (length) of exactly 1
• The projection shows how much of vector A points in the direction of vector B