Degrees Between Two Vectors Online Calculator
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Reviewed by Calculator Editorial Team
This calculator helps you find the angle between two vectors in degrees. Whether you're studying physics, engineering, or computer graphics, understanding the angle between vectors is essential for many calculations.
How to Use This Calculator
To calculate the angle between two vectors, follow these simple steps:
- Enter the components of the first vector (x₁ and y₁).
- Enter the components of the second vector (x₂ and y₂).
- Click the "Calculate" button to get the angle in degrees.
- The result will show the angle between the two vectors in degrees.
The calculator uses the dot product formula to determine the angle between the vectors. This method is reliable and widely used in physics and mathematics.
Formula Explained
The angle θ between two vectors A and B can be calculated using the dot product formula:
Dot Product Formula
A · B = |A| |B| cosθ
Where:
- A and B are the vector components
- |A| and |B| are the magnitudes of the vectors
- θ is the angle between the vectors
To find θ, we rearrange the formula:
Angle Calculation
θ = arccos[(A · B) / (|A| |B|)]
This formula gives us the angle in radians, which we then convert to degrees by multiplying by 180/π.
Worked Example
Let's calculate the angle between two vectors with components (3, 4) and (1, 2).
- Calculate the dot product: (3)(1) + (4)(2) = 3 + 8 = 11
- Calculate the magnitudes: |A| = √(3² + 4²) = 5, |B| = √(1² + 2²) ≈ 2.236
- Calculate the cosine of the angle: cosθ = 11 / (5 × 2.236) ≈ 0.982
- Find the angle in radians: θ ≈ arccos(0.982) ≈ 0.182 radians
- Convert to degrees: θ ≈ 0.182 × (180/π) ≈ 10.4°
The angle between the vectors (3, 4) and (1, 2) is approximately 10.4 degrees.
Frequently Asked Questions
What is the angle between two vectors?
The angle between two vectors is the smallest angle formed when the two vectors are placed tail-to-tail. It's measured in degrees or radians.
How do I calculate the angle between two vectors?
You can calculate the angle using the dot product formula, which involves finding the dot product of the vectors, their magnitudes, and then using the arccosine function.
What if the angle is greater than 90 degrees?
The dot product formula will give you the smallest angle between the vectors. If you need the larger angle, you can subtract the result from 180 degrees.
Can I use this calculator for 3D vectors?
This calculator is designed for 2D vectors. For 3D vectors, you would need to extend the formula to include the z-component.