Angle Between Velocity and Acceleration Vectors Calculator

Formula

The formula to calculate the angle between the velocity and acceleration vectors is:

\[ A = \cos^{-1} \left( \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{A}|| \cdot ||\mathbf{B}||} \right) \]

Where:

\(A\) is the angle between the vectors (degrees)
\(\mathbf{a} \cdot \mathbf{b}\) is the dot product of the two vectors
\(||\mathbf{A}||\) is the magnitude of vector \(\mathbf{A}\)
\(||\mathbf{B}||\) is the magnitude of vector \(\mathbf{B}\)

What are Velocity and Acceleration Vectors?

Velocity and acceleration vectors are 3-Dimensional representations of the physical properties of velocity and acceleration. The velocity vector represents the speed and direction of an object, while the acceleration vector represents the rate of change of the velocity vector.

Example Calculation

Let's assume the following values:

Velocity Vector: v1 = (3, 4, 0)
Acceleration Vector: a1 = (1, 2, 2)

Using the formula:

\[ \mathbf{a} \cdot \mathbf{b} = (3 \cdot 1) + (4 \cdot 2) + (0 \cdot 2) = 3 + 8 + 0 = 11 \] \[ ||\mathbf{A}|| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16 + 0} = \sqrt{25} = 5 \] \[ ||\mathbf{B}|| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \] \[ A = \cos^{-1} \left( \frac{11}{5 \cdot 3} \right) = \cos^{-1} \left( \frac{11}{15} \right) = 42.27 \text{ degrees} \]

The angle between the vectors is 42.27 degrees.