Coordinate Angle Calculator

Formula

To calculate the angle between two vectors (\(θ\)):

\[ θ = \arccos\left(\frac{v1_x \cdot v2_x + v1_y \cdot v2_y}{|v1| \cdot |v2|}\right) \]

Where:

\(θ\) is the angle between the vectors in degrees
\(v1_x\) and \(v1_y\) are the x and y components of vector 1
\(v2_x\) and \(v2_y\) are the x and y components of vector 2
\(|v1|\) is the magnitude of vector 1
\(|v2|\) is the magnitude of vector 2

What is a Coordinate Angle?

A coordinate angle is the angle formed between two vectors in a coordinate system. This angle is crucial in various fields such as physics, engineering, and computer graphics, as it helps in understanding the spatial relationship between different vectors. The angle can be calculated using the dot product of the vectors and their magnitudes, providing insights into their direction and orientation.

Example Calculation 1

Let's assume the following values:

Vector 1 (\(v1\)): \(v1_x = 3\), \(v1_y = 4\)
Vector 2 (\(v2\)): \(v2_x = 4\), \(v2_y = 3\)

Using the formula:

\[ θ = \arccos\left(\frac{3 \cdot 4 + 4 \cdot 3}{\sqrt{3^2 + 4^2} \cdot \sqrt{4^2 + 3^2}}\right) = \arccos\left(\frac{24}{25}\right) \approx 16.26^\circ \]

The angle between the vectors is approximately 16.26 degrees.

Example Calculation 2

Let's assume the following values:

Vector 1 (\(v1\)): \(v1_x = 1\), \(v1_y = 0\)
Vector 2 (\(v2\)): \(v2_x = 0\), \(v2_y = 1\)

Using the formula:

\[ θ = \arccos\left(\frac{1 \cdot 0 + 0 \cdot 1}{\sqrt{1^2 + 0^2} \cdot \sqrt{0^2 + 1^2}}\right) = \arccos(0) = 90^\circ \]

The angle between the vectors is 90 degrees.