3D Vector Dot Product, Cross Product & Angle Calculator

Evaluate 3D vector dot product, enclosed angle, and parallelogram area |A×B|.

Scalar Dot Product (A · B)
12
Angle Between Vectors (θ) 61.315 deg
Cross Product Magnitude |A × B| 21.932
Magnitude |A| 5
Magnitude |B| 5
Inputs
Vector A x-component.
Vector A y-component.
Vector A z-component.
Vector B x-component.
Vector B y-component.
Vector B z-component.
Results & Graph
Scalar Dot Product (A · B)
12
Angle Between Vectors (θ)
61.315 deg
Cross Product Magnitude |A × B|
21.932
Magnitude |A|
5
Magnitude |B|
5

Scalar Dot Product (A · B)

Graph 1 • Ax Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 9.7 19.4 50%: 6 6 50% 75%: 9 9 75% 100%: 12 12 100% 125%: 15 15 125% 150%: 18 18 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Scalar Dot Product (A · B) 12 Angle Between Vectors (θ) 61.315 deg Cross Product Magnitude |A × B| 21.932 Magnitude |A| 5
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Scalar Dot Product (A · B)12Angle Between Vectors (θ)61.315 degCross Product Magnitude |A × B|21.932Magnitude |A|5Magnitude |B|5
Summary: A·B = 12, angle θ = 61.31°, and cross-product magnitude |A × B| = 21.9317.
Summary: A·B = 12, angle θ = 61.31°, and cross-product magnitude |A × B| = 21.9317.

Formula & Mathematical Methodology: 3D Vector Math Mathematical Identity

A·B = axbx + ayby + azbz
cos(θ) = A·B|A||B|
A·B = a_xb_x + a_yb_y + a_zb_z; cos(θ) = (A·B) / (|A||B|)
  • ax: Ax (numeric parameter)
  • ay: Ay (numeric parameter)
  • az: Az (numeric parameter)
  • bx: Bx (numeric parameter)
  • by: By (numeric parameter)
  • bz: Bz (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: 3D Vector Math

Evaluate 3D vector dot product, enclosed angle, and parallelogram area |A×B|.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: A·B = a_xb_x + a_yb_y + a_zb_z; cos(θ) = (A·B) / (|A||B|).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates 3D Vector Math via A·B = a_xb_x + a_yb_y + a_zb_z; cos(θ) = (A·B) / (|A||B|).

Frequently Asked Questions

What mathematical formula does the 3D Vector Dot Product, Cross Product & Angle Calculator use?

It directly evaluates A·B = a_xb_x + a_yb_y + a_zb_z; cos(θ) = (A·B) / (|A||B|) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Coordinate & Vector Geometry Guide

How to Use the 3D Vector Dot Product, Cross Product & Angle Calculator: Practical Guide & Visual Analysis

Calculate 3D vector magnitudes |A| and |B|, scalar dot product A·B, angle θ between vectors, and cross-product magnitude |A×B|. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching 3d vector dot product, cross product & angle calculator, 3d vector math, vector 3d dot cross product angle, coordinate & vector geometry calculator, this interactive tool helps you calculate 3D vector magnitudes |A| and |B|, scalar dot product A·B, angle θ between vectors, and cross-product magnitude |A×B|.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Ax, Ay, Az, Bx, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: 3D Vector Math), Evaluates 3D Vector Math via A·B = a_xb_x + a_yb_y + a_zb_z; cos(θ) = (A·B) / (|A||B|). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).