Vector Calculator - Add, Subtract, Dot Product, Cross Product

What vector operations are and when to use them

A vector represents a quantity with both magnitude and direction — velocity, force, displacement, or any directional quantity in physics, engineering, computer graphics, and navigation. This calculator performs the core operations you need when working with 2D or 3D vectors: addition and subtraction (combining or comparing directional quantities), the dot product (a scalar measuring how much two vectors point in the same direction), the cross product (a vector perpendicular to both inputs, used in 3D for torque, normals, and rotation), and magnitude (a vector's length).

Use it for physics coursework (combining forces or velocities), computer graphics and game development (surface normals, lighting calculations), robotics and navigation (relative positions and headings), or any problem where you need to combine, compare, or measure directional quantities. Enter only the i (x) and j (y) components for a 2D vector, or all three for a 3D vector — the calculator automatically switches to 3D mode if either z-component is nonzero.

The formulas

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

• Addition: A + B = (a₁+b₁, a₂+b₂, a₃+b₃)
• Subtraction: A − B = (a₁−b₁, a₂−b₂, a₃−b₃)
• Dot product: A · B = a₁b₁ + a₂b₂ + a₃b₃ (a single number, not a vector)
• Cross product (3D): A × B = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁) — a vector perpendicular to both A and B
• Magnitude: |A| = √(a₁² + a₂² + a₃²)
• Angle between vectors: θ = cos⁻¹(A · B ÷ (|A| × |B|))

Worked example

Let Vector A = (3, 4) and Vector B = (1, 2) — both 2D, so z-components are left blank.

• A + B = (3+1, 4+2) = (4, 6)
• A − B = (3−1, 4−2) = (2, 2)
• A · B = (3×1) + (4×2) = 3 + 8 = 11
• |A| = √(3² + 4²) = √25 = 5; |B| = √(1² + 2²) = √5 ≈ 2.2361
• Angle = cos⁻¹(11 ÷ (5 × 2.2361)) = cos⁻¹(0.9839) ≈ 10.3048°

This matches the calculator's output for those inputs, including the 2D cross product's z-component: (3×2) − (4×1) = 2.

Common mistakes / how to interpret

• Confusing dot and cross products. The dot product returns a single number (scalar); the cross product returns another vector. Mixing them up in a formula produces nonsensical units or dimensions.
• Computing a 3D cross product from 2D vectors incorrectly. Two purely 2D vectors technically have a cross product that points entirely in the z-direction (out of the plane) — this calculator reports that single z-component rather than a full 3-component vector when both inputs are 2D.
• Forgetting the dot product depends on the angle between vectors. A dot product of zero means the vectors are perpendicular (90°), a positive dot product means the angle is less than 90°, and a negative dot product means the angle is greater than 90° — the sign alone tells you useful information before you even compute the angle.
• Mixing units between vectors. Adding or comparing vectors measured in different units (e.g., meters and feet) without converting first will silently produce a meaningless result.

Frequently Asked Questions

Common Tip Percentages

General Household Budget Ratios (50/30/20 Rule)

Common Measurement Conversions for Everyday Use