Calculate the angle between two 2D or 3D vectors using the dot product formula. Includes magnitudes, dot product, cross product, projection, parallel/perpendicular detection, and visual gauge.
The angle between two vectors is a fundamental concept in linear algebra, physics, computer graphics, and machine learning. It measures how much two directions diverge, from 0° (perfectly aligned) through 90° (perpendicular) to 180° (opposite). The standard formula uses the dot product: cos θ = (A · B) / (|A| |B|).
This calculator handles both 2D and 3D vectors. Enter the components of each vector and instantly see the angle in degrees and radians, the dot product, the cross product (full 3D vector or scalar z-component for 2D), the magnitude of each vector, the cosine of the angle, and the scalar and vector projections.
Parallel and perpendicular relationships are detected automatically. Parallel vectors (0° or 180°) have a dot product equal to ±|A||B|, while perpendicular vectors have a dot product of exactly zero. These checks are essential in physics (force decomposition), computer graphics (normal testing), and data science (cosine similarity).
The visual angle gauge provides an instant intuitive sense of where your angle falls on the 0°–180° spectrum. The magnitude comparison bar chart shows |A|, |B|, |A+B|, and |A−B|, illustrating the triangle inequality and parallelogram law. A reference table of notable angles (0°, 30°, 45°, 60°, 90°, etc.) helps put your result in context.
Presets demonstrate classic cases — orthogonal basis vectors, 45° diagonals, parallel and anti-parallel pairs — making it easy to build intuition or verify your understanding.
This angle between vectors calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Dimension, Ax, Ay, Az and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
cos θ = (A · B) / (|A| · |B|) Dot product: A · B = Ax·Bx + Ay·By + Az·Bz Cross product (3D): A × B = (AyBz − AzBy, AzBx − AxBz, AxBy − AyBx) |V| = √(Vx² + Vy² + Vz²) Projection of A onto B: proj = (A·B / |B|²) · B Scalar projection: comp = A·B / |B|
Result: For dim=2d, ax=1, ay=0, the tool returns the solved angle between vectors outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in angle between vectors formulas and reports derived values, checks, and classifications automatically.
This page is tailored to angle between vectors, with outputs tied directly to the form fields (Dimension, Ax, Ay, Az). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
Use the dot product formula: θ = arccos((A·B) / (|A|·|B|)). Compute the dot product, divide by the product of magnitudes, and take the inverse cosine.
It means the vectors are perpendicular (orthogonal). The angle between them is exactly 90°.
In 3D, the cross product A×B is a vector perpendicular to both A and B, with magnitude |A||B|sin θ. In 2D, it simplifies to a scalar (the z-component).
No — the angle between two vectors is always between 0° and 180° (inclusive). The arccos function naturally returns values in this range.
Two vectors are parallel if their cross product is zero (or the angle is 0° or 180°). Equivalently, one is a scalar multiple of the other.
Cosine similarity is cos θ = (A·B)/(|A||B|), the same formula this calculator uses. It ranges from −1 (opposite) through 0 (perpendicular) to 1 (identical direction), and is widely used in NLP, recommendation systems, and clustering.