Angle Between Vectors Calculator Degrees

Calculate the angle between two vectors in degrees with our precise vector angle calculator. This tool helps you determine the angle formed by two vectors in 2D or 3D space, which is essential for physics, engineering, and computer graphics applications.

How to Use This Calculator

Using our angle between vectors calculator is simple:

Enter the components of the first vector (x₁, y₁, z₁)
Enter the components of the second vector (x₂, y₂, z₂)
Click "Calculate" to get the angle in degrees
View the result and visualization

The calculator will display the angle between the vectors in degrees, along with a visualization of the vectors and their angle.

Formula Explained

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

A · B = |A| |B| cosθ

θ = arccos[(A · B) / (|A| |B|)]

Where:

A and B are the vector components
A · B is the dot product of vectors A and B
|A| and |B| are the magnitudes of vectors A and B

The result is converted from radians to degrees for easier interpretation.

Worked Examples

Let's look at a couple of examples to understand how the angle between vectors is calculated.

Example 1: 2D Vectors

Given vectors A = (3, 4) and B = (5, 12):

A · B = (3×5) + (4×12) = 15 + 48 = 63

|A| = √(3² + 4²) = √(9 + 16) = √25 = 5

|B| = √(5² + 12²) = √(25 + 144) = √169 = 13

θ = arccos(63 / (5×13)) = arccos(63/65) ≈ 0.4636 radians ≈ 26.565°

Example 2: 3D Vectors

Given vectors A = (1, 2, 3) and B = (4, 5, 6):

A · B = (1×4) + (2×5) + (3×6) = 4 + 10 + 18 = 32

|A| = √(1² + 2² + 3²) = √(1 + 4 + 9) = √14 ≈ 3.7417

|B| = √(4² + 5² + 6²) = √(16 + 25 + 36) = √77 ≈ 8.7750

θ = arccos(32 / (3.7417×8.7750)) ≈ arccos(32/32.83) ≈ arccos(0.9746) ≈ 0.2257 radians ≈ 13.008°

Example Table

Vector A	Vector B	Angle (degrees)
(1, 0)	(0, 1)	90.000°
(1, 1)	(-1, 1)	90.000°
(3, 4)	(4, 3)	18.4349°

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.
How do I calculate the angle between two vectors?: You can calculate the angle using the dot product formula: θ = arccos[(A · B) / (|A| |B|)], then convert the result from radians to degrees.
What is the difference between 2D and 3D vector angles?: The calculation method is the same for both 2D and 3D vectors, but 3D vectors have an additional z-component to consider in the calculations.
Can the angle between vectors be greater than 180 degrees?: No, the angle between vectors is always between 0 and 180 degrees, representing the smallest angle between the two vectors.
How accurate is this calculator?: This calculator uses precise mathematical calculations and provides results with up to 4 decimal places for accuracy.