Angle Between Vectors Calculator
Angle Between Two Vectors works out Angle and Angle in the chosen unit from Vector a (components, comma separated), Vector b (components, comma separated) and aₓ. The default figures on this page give angle of 16.260205 deg; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
7 steps
- a·b = (3)(4) + (4)(3) = 24
- |a| = √(3² + 4²) = 5
- |b| = √(4² + 3²) = 5
- cos θ = (a·b) / (|a||b|) = 24 / (5 × 5) = 0.96
- θ = arccos(0.96) = 16.26020471° = 0.2837941092 rad
- a × b = (0, 0, -7), |a × b| = 7
- Scalar projection of a onto b = (a·b)/|b| = 4.8
Cross product a × btable
Vector projection of a onto btable
Unit vector âtable
Unit vector b̂table
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
What the Angle Between Two Vectors calculates
Finds the angle between two vectors of any dimension from the dot product formula cos θ = (a·b)/(|a||b|), and returns the dot product, magnitudes, cosine similarity, cross product, projection and whether the vectors are perpendicular or parallel.
How to use this Angle Between Two Vectors
Enter Vector a (components, comma separated), Vector b (components, comma separated), aₓ and a_y. Angle updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Angle, Angle in the chosen unit and Dot product a·b.
The method follows Weisstein, Eric W. "Dot Product." MathWorld — a·b = |a||b| cos θ, Weisstein, Eric W. "Cross Product." MathWorld — |a × b| = |a||b| sin θ and Weisstein, Eric W. "Vector Projection." MathWorld.
Perpendicular unit vectors i and j example
With Vector a (components, comma separated): 1,0, Vector b (components, comma separated): 0,1, the calculator returns Angle: 90, Dot product a·b: 0, cos θ (cosine similarity): 0. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Angle, Angle in the chosen unit and Dot product a·b. The inputs you can vary are Vector a (components, comma separated), Vector b (components, comma separated), aₓ, a_y, a_z and bₓ.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
What is the formula for angle between?
On the figures this page opens with, the angle is 16.260205 deg. No single number fits everyone: it follows Vector a (components, comma separated), Vector b (components, comma separated) and aₓ, so enter your own above and the answer updates as you type.
What is the formula for calculating the angle between two unit vectors?
Finds the angle between two vectors of any dimension from the dot product formula cos θ = (a·b)/(|a||b|), and returns the dot product, magnitudes, cosine similarity, cross product. On this page it is computed from Vector a (components, comma separated), Vector b (components, comma separated) and aₓ and shown as Angle, Angle in the chosen unit and Dot product a·b.
How do I calculate the angle between two points?
Finds the angle between two vectors of any dimension from the dot product formula cos θ = (a·b)/(|a||b|), and returns the dot product, magnitudes, cosine similarity, cross product. Open “How this is worked out” beside the result to see each step for your own figures.
What is the formula for finding the angle?
It needs Vector a (components, comma separated), Vector b (components, comma separated), aₓ, a_y, a_z and bₓ. Nothing else is required, and the figures stay in your browser.
What information do I need for the Angle Between Two Vectors?
You need Vector a (components, comma separated), Vector b (components, comma separated), aₓ, a_y, a_z, bₓ, b_y and b_z. The page opens with default figures that give angle of 16.260205 deg, so you can see the scale before entering your own.
What does this Angle Between Two Vectors show?
It shows Angle, Angle in the chosen unit, Dot product a·b, |a|, |b|, cos θ (cosine similarity), Cross product a × b and |a × b|, with the working and any warnings for your entries.