Quick answer
The angle between vectors a and b is θ = arccos((a · b) ÷ (|a||b|)). For a = (1, 0) and b = (0, 1) the dot product is 0, so θ = 90° (1.5708 rad).
Enter the components of each vector, separated by commas. The calculator returns the angle in degrees and radians. Use this angle between two vectors calculator to check your own figures.
Related tools: Line Equation from Two Points Calculator: Equation of the line through two points. Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers.
How to use the Angle Between Two Vectors Calculator
- Type vector A, for example 1, 2, 3.
- Type vector B, for example 1, 2, 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Vector A | Text, such as 1, 2, 3 | — |
| Vector B | Text, such as 1, 2, 3 | — |
Formula and method
Formula
θ = arccos((a · b) ÷ (|a||b|))
IEEE-754 double precision; shown to at most 6 decimal places.
Uses the numerically stable form θ = 2·atan2(|â − b̂|, |â + b̂|) so parallel and opposite vectors give exactly 0° and 180°. Neither vector may be zero.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Vector A = 1, 0; Vector B = 0, 1.
| Result | Value |
|---|---|
| Angle | 90° |
| Radians | 1.570796 |
Example 2
Inputs: Vector A = 1, 0; Vector B = 1, 1.
| Result | Value |
|---|---|
| Angle | 45° |
| Radians | 0.785398 |
Example 3
Inputs: Vector A = 1, 2; Vector B = -1, -2.
| Result | Value |
|---|---|
| Angle | 180° |
| Radians | 3.141593 |
The first line is the headline result (angle). Below it the angle Between Two Vectors Calculator lists radians, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Reading the dot product
A positive dot product means an acute angle, zero means perpendicular and negative means obtuse. For a = (3, 4) and b = (4, 3): a·b = 24, |a||b| = 25, so θ = arccos(0.96) = 16.26°.
Requirements
Vectors must be non-zero and have the same number of components. The same formula works in two, three or more dimensions.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Vector A: entering "0, 0" shows "Vectors must be non-zero." beside that field and no result.
Choosing between the Angle Between Two Vectors Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Line Equation from Two Points Calculator](/math/line-equation-from-two-points/) | x₁, y₁ and x₂ | Equation |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Non-zero vectors
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for vector A to confirm this.
Frequently asked questions
What if the vectors are parallel?
The angle is 0° or 180°.
What is the dot product?
The sum of products of components.
Can I use 3D vectors?
Yes, enter three components.
What does the Angle Between Two Vectors Calculator calculate?
Angle between two vectors. It returns Angle and Radians.
What formula does the Angle Between Two Vectors Calculator use?
θ = arccos((a · b) ÷ (|a||b|)). Uses the numerically stable form θ = 2·atan2(|â − b̂|, |â + b̂|) so parallel and opposite vectors give exactly 0° and 180°.
What do I need to enter in the angle Between Two Vectors Calculator?
You enter Vector A and Vector B.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Non-zero vectors.
What is an example of a angle Between Two Vectors result?
With vector A 1, 0 and vector B 0, 1, the angle is 90°.
Related calculators
- Line Equation from Two Points Calculator — Equation of the line through two points.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
- Prime Factorization Calculator — Break a whole number into its prime factors.
- Exponent Calculator — Raise a base to any power.
- Perfect Square Calculator — Check whether a number is a perfect square.
Sources
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.