Angle Between Vectors Calculator

Use our free angle between vectors calculator to compute the angle θ between two vectors, cos(θ), check orthogonality and parallelism, and classify the relationship.

Math

Mode

𝐚 =

𝐛 =

Formula

Compute the angle θ between two vectors using the dot product formula. Get the angle in radians and degrees, cos(θ), and classification (acute, obtuse, right, or parallel). Also check orthogonality and compute cosine similarity.

θ = arccos((𝐚·𝐛)/(|𝐚||𝐛|))

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Enter 𝐚 and 𝐛 as comma-separated numbers (same dimension).
2Select mode: compute angle, orthogonality check, or cosine similarity.
3Click Calculate.

Examples

Angle between (1,0,0) and (0,1,0)

1.𝐚·𝐛 = 0, |𝐚|=1, |𝐛|=1
2.cos(θ) = 0, θ = π/2 = 90°

Result: 1.5707963268

Orthogonality of (1,1) and (−1,1)

1.𝐚·𝐛 = −1+1 = 0 ✓ orthogonal

Result: 1

Frequently Asked Questions

θ = arccos((𝐚·𝐛)/(|𝐚||𝐛|)) using the dot product formula.

cos(θ)=1 → parallel, 0 → orthogonal, −1 → anti-parallel.

Yes — works for 2D, 3D, and higher dimensions.

The angle is undefined — zero vectors have no direction.

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CalculateMe Team

Last updated: 2026-07-15

Angle Between Vectors Calculator

Formula

How It Works

Examples

Angle between (1,0,0) and (0,1,0)

Orthogonality of (1,1) and (−1,1)

Frequently Asked Questions

How is the angle computed?

What does cos(θ) tell us?

Can I use any dimension?

What if a vector is zero?

Related Calculators

Angle Between Vectors Calculator

Formula

How It Works

Examples

Angle between (1,0,0) and (0,1,0)

Orthogonality of (1,1) and (−1,1)

Frequently Asked Questions

How is the angle computed?

What does cos(θ) tell us?

Can I use any dimension?

What if a vector is zero?

Related Calculators