📐 Angle Between Two Vectors Calculator
Calculate the angle between two vectors in 2D or 3D space
Vector 1 (u)
Vector 2 (v)
How to Use This Calculator
Enter Vector Components
Input the x, y (and optionally z) components of the first vector. For 2D vectors, leave z as 0 or empty.
Enter Second Vector
Input the x, y (and optionally z) components of the second vector.
Calculate
Click "Calculate Angle" to find the angle between the two vectors in degrees and radians.
Formula
cos(θ) = (u · v) / (|u| × |v|)
θ = arccos((u · v) / (|u| × |v|))
Where:
- u · v = Dot product of vectors u and v
- |u| = Magnitude (length) of vector u = √(x₁² + y₁² + z₁²)
- |v| = Magnitude (length) of vector v = √(x₂² + y₂² + z₂²)
- θ = Angle between the vectors (in radians or degrees)
Dot Product Formula:
u · v = x₁x₂ + y₁y₂ + z₁z₂
Example 1: Find the angle between u = (1, 2) and v = (3, 4)
u · v = (1)(3) + (2)(4) = 3 + 8 = 11
|u| = √(1² + 2²) = √5 ≈ 2.236
|v| = √(3² + 4²) = √25 = 5
cos(θ) = 11 / (2.236 × 5) = 11 / 11.18 ≈ 0.984
θ = arccos(0.984) ≈ 10.3°
Example 2: Find the angle between u = (1, 0, 0) and v = (0, 1, 0)
u · v = (1)(0) + (0)(1) + (0)(0) = 0
|u| = |v| = 1
cos(θ) = 0 / (1 × 1) = 0
θ = arccos(0) = 90° (perpendicular vectors)
About Angle Between Two Vectors Calculator
The Angle Between Two Vectors Calculator finds the angle between two vectors in 2D or 3D space using the dot product formula. This angle represents the smallest angle through which one vector must rotate to align with the other vector.
When to Use This Calculator
- Linear Algebra: Calculate angles between vectors
- Physics: Find angles between force vectors, velocity vectors, or field vectors
- Computer Graphics: Determine angles for rotations and transformations
- Engineering: Calculate angles in mechanical systems and structural analysis
- Geometry: Find angles between lines or directions
- Navigation: Calculate angles between directions or bearings
Why Use Our Calculator?
- ✅ 2D & 3D Support: Works with vectors in any dimension
- ✅ Instant Results: Calculate angle immediately
- ✅ Detailed Steps: Shows dot product, magnitudes, and cosine value
- ✅ Degrees & Radians: Provides results in both units
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Vector Angles
The angle between two vectors is always between 0° and 180° (0 and π radians). Key angle values:
- 0°: Vectors point in the same direction (parallel and same orientation)
- 90°: Vectors are perpendicular (orthogonal)
- 180°: Vectors point in opposite directions (parallel but opposite orientation)
- 0° < θ < 90°: Acute angle (vectors point in similar directions)
- 90° < θ < 180°: Obtuse angle (vectors point in different directions)
Real-World Applications
Physics: Finding the angle between force vectors helps determine the resultant force and work done.
Computer Graphics: Calculating angles between surface normals determines lighting and shading effects.
Robotics: Determining angles between joint vectors is essential for inverse kinematics.
Frequently Asked Questions
What is the angle between two vectors?
The angle between two vectors is the smallest angle through which one vector must rotate to align with the other. It ranges from 0° to 180°.
How is the angle calculated?
The angle is calculated using the dot product formula: cos(θ) = (u·v) / (|u|×|v|), then θ = arccos(...). This uses the dot product and magnitudes of both vectors.
Can I use this for 2D vectors?
Yes! Simply enter x and y components, and leave z as 0 or empty. The calculator works for both 2D and 3D vectors.
What does a 90° angle mean?
A 90° angle means the vectors are perpendicular (orthogonal). Their dot product equals zero, indicating no component of one vector lies in the direction of the other.
What if the vectors are parallel?
If vectors point in the same direction, the angle is 0°. If they point in opposite directions, the angle is 180°. In both cases, they are parallel (collinear).
Why use the dot product?
The dot product relates vector magnitudes and the angle between them. It's the most efficient way to calculate angles without complex geometry, especially in higher dimensions.
What happens if a vector has zero magnitude?
A zero vector has no direction, so the angle is undefined. The calculator will show an error if you enter a zero vector (0, 0, 0).
Can the angle be negative?
No, the angle between two vectors is always between 0° and 180° (0 and π radians). The calculator finds the smallest angle, which is never negative.