Math
Angle Between Two Vectors Calculator
Enter two vectors and mathcheck computes the angle between them from cos θ = (a · b) / (|a| |b|).
You get degrees and radians (exact multiples of π when the angle is a special angle), the cosine in rationalized exact form, both magnitudes, and a note on whether the vectors are perpendicular, parallel, acute, or obtuse.
Horizontal component of Vector a.
Vertical component of Vector a.
Horizontal component of Vector b.
Vertical component of Vector b.
Try an example
Result
Angle θ
—
- Degrees
- —
- Radians
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- cos θ
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- Relationship
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- Supplementary angle (180° - θ)
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Student quick launch
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Study path
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The angle between vectors
Place two A quantity with both a size and a direction, often drawn as an arrow. tail to tail; the angle between them is the smaller angle from one to the other, from 0° (same direction) to 180° (opposite directions). The Multiply two vectors piece by piece and add the results to get one number. makes it easy to find: divide a · b by the product of the The length or size of a vector, ignoring which way it points. to get cos θ, then take the inverse cosine.
a = ⟨1, 0⟩ and b = ⟨1, 1⟩
a · b = 1. |a| = 1, |b| = √2. cos θ = 1/√2 = √2/2, so θ = 45° = π/4 rad.
Special angles you can recognize
| cos θ | θ (degrees) | θ (radians) |
|---|---|---|
| 1 | 0° | 0 |
| √3/2 | 30° | π/6 |
| √2/2 | 45° | π/4 |
| 1/2 | 60° | π/3 |
| 0 | 90° | π/2 |
| -1/2 | 120° | 2π/3 |
| -1 | 180° | π |
How to use this calculator
- Choose 2D or 3D.
- Enter the components of both vectors.
- Read θ in degrees with the An angle unit where a full turn is 2π instead of 360 degrees. form beside it; special angles appear as exact multiples of π.
- Check the relationship badge to see whether the vectors are perpendicular, parallel, acute, or obtuse.
- Open Show the work for the dot product, magnitudes, and the arccos step.
How to read the answer
An angle of 0° means the vectors point the same way, 90° means they are perpendicular, and 180° means they point in opposite directions. Angles under 90° come from a positive dot product, angles over 90° from a negative one. The cosine value is often a cleaner exact answer than the angle itself, so the calculator gives it in rationalized form such as √2/2.
Common mistakes and edge cases
- Using the tangent formula for lines. Vectors have direction, so their angle can exceed 90°; the dot-product formula handles that.
- Forgetting to divide by both magnitudes.
- Rounding the cosine before taking arccos, which shifts the angle by a fraction of a degree.
- Trying to find the angle with a zero vector; it has no direction so the angle is undefined.
- Reporting the answer in radians when the question asks for degrees (or the reverse).
Worked examples
45° special angle
⟨1, 0⟩ and ⟨1, 1⟩
Angle θ
45° (π/4 rad)
Perpendicular vectors
⟨3, 4⟩ and ⟨-4, 3⟩
Angle θ
90° (π/2 rad)
Obtuse angle
⟨1, 2⟩ and ⟨-3, 1⟩
Angle θ
98.13° (1.7127 rad)
3D vectors
⟨1, 2, 3⟩ and ⟨4, -5, 6⟩
Angle θ
68.56° (1.1966 rad)
Opposite directions
⟨2, 2⟩ and ⟨-1, -1⟩
Angle θ
180° (3.1416 rad)
Decimal components
⟨1.5, 0.5⟩ and ⟨0.5, 1.5⟩
Angle θ
53.13° (0.9273 rad)
Zero vector
Angle θ
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Frequently asked questions
What is the formula for the angle between two vectors?+
cos θ = (a · b) / (|a| |b|). Compute the dot product, divide by the product of the two magnitudes, and take the inverse cosine.
Can the angle between two vectors be more than 180°?+
No. The angle between vectors is defined as the smaller angle between their directions, so it is always between 0° and 180°.
How do I know if two vectors are perpendicular?+
Their dot product is 0, which makes cos θ = 0 and θ = 90°. For example, ⟨3, 4⟩ · ⟨-4, 3⟩ = -12 + 12 = 0.
How do I convert the angle to radians?+
Multiply degrees by π/180. The calculator shows special angles as exact fractions of π, such as 60° = π/3.
Does the formula work in 3D?+
Yes. The dot product and magnitudes just include the z-components. The angle is measured in the plane containing both vectors.
Why is my cosine value slightly off from a special value?+
Decimal components introduce rounding. With integer components the calculator computes the cosine exactly and rationalizes it, such as √2/2 instead of 0.7071.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026