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Angle Between Two Vectors Calculator Degrees

Reviewed by Calculator Editorial Team

Calculating the angle between two vectors is essential in physics, engineering, and computer graphics. This calculator provides an accurate measurement in degrees, along with a visual representation of the vectors and their angle.

How to Use This Calculator

To calculate the angle between two vectors:

  1. Enter the components of the first vector (x₁ and y₁)
  2. Enter the components of the second vector (x₂ and y₂)
  3. Click "Calculate" to see the angle in degrees
  4. View the visualization to understand the geometric relationship

The calculator uses the dot product formula to determine the angle between the vectors. The result is displayed in degrees with two decimal places of precision.

The Formula Explained

The angle θ between two vectors A and B can be calculated using the dot product formula:

θ = arccos[(A · B) / (||A|| ||B||)]

Where:

  • A · B is the dot product of vectors A and B
  • ||A|| is the magnitude of vector A
  • ||B|| is the magnitude of vector B

The dot product A · B is calculated as:

A · B = (x₁ * x₂) + (y₁ * y₂)

The magnitudes of the vectors are calculated as:

||A|| = √(x₁² + y₁²)

||B|| = √(x₂² + y₂²)

The result is converted from radians to degrees by multiplying by 180/π.

Worked Examples

Example 1: Simple Vectors

Given vectors A = (1, 0) and B = (0, 1):

  1. Calculate dot product: A · B = (1*0) + (0*1) = 0
  2. Calculate magnitudes: ||A|| = √(1² + 0²) = 1, ||B|| = √(0² + 1²) = 1
  3. Calculate angle: θ = arccos(0 / (1*1)) = arccos(0) = 90°

The angle between these perpendicular vectors is 90 degrees.

Example 2: Non-Orthogonal Vectors

Given vectors A = (3, 4) and B = (1, 2):

  1. Calculate dot product: A · B = (3*1) + (4*2) = 3 + 8 = 11
  2. Calculate magnitudes: ||A|| = √(3² + 4²) = 5, ||B|| = √(1² + 2²) = √5
  3. Calculate angle: θ = arccos(11 / (5*√5)) ≈ 26.565°

The angle between these vectors is approximately 26.57 degrees.

Visualization

The calculator includes a visualization that shows the two vectors and their angle. This helps you understand the geometric relationship between the vectors.

Note: The visualization is interactive and updates when you change the vector components.

Frequently Asked Questions

What is the angle between two vectors?
The angle between two vectors is the smallest angle formed when the vectors are placed tail-to-tail. It's calculated using the dot product formula.
Can the angle between vectors be greater than 180 degrees?
No, the angle between vectors is always between 0 and 180 degrees. The formula automatically gives the smallest angle between the vectors.
What happens if the vectors are parallel?
If the vectors are parallel, the angle will be 0 degrees if they point in the same direction, or 180 degrees if they point in opposite directions.
How accurate is this calculator?
This calculator uses precise mathematical formulas and provides results with two decimal places of accuracy.