Angle Between Two Vectors Calculator

How to Calculate the Angle Between Two Vectors

Calculating the angle between two vectors is a fundamental operation in linear algebra and has numerous applications in physics, engineering, and computer graphics. This calculation involves using the dot product of the vectors and their magnitudes. Here's a comprehensive guide on how to calculate the angle between two vectors:

Angle Between Vectors Formula

The formula to calculate the angle between two vectors \(\vec{a}\) and \(\vec{b}\) is:

\[ \theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right) \]

Where:

• \(\theta\) is the angle between the vectors
• \(\vec{a} \cdot \vec{b}\) is the dot product of the vectors
• \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes (lengths) of the vectors
• \(\arccos\) is the inverse cosine function

Calculation Steps

1. Calculate the dot product of the two vectors
2. Calculate the magnitude of each vector
3. Divide the dot product by the product of the magnitudes
4. Take the inverse cosine (arccos) of the result
5. Convert the angle from radians to degrees if needed

Example Calculation (2D)

Let's calculate the angle between two 2D vectors:

\(\vec{a} = (1, 2)\) and \(\vec{b} = (4, 5)\)

1. Calculate the dot product: \[ \vec{a} \cdot \vec{b} = (1 \times 4) + (2 \times 5) = 4 + 10 = 14 \]
2. Calculate the magnitudes: \[ |\vec{a}| = \sqrt{1^2 + 2^2} = \sqrt{5} \approx 2.236 \] \[ |\vec{b}| = \sqrt{4^2 + 5^2} = \sqrt{41} \approx 6.403 \]
3. Apply the formula: \[ \theta = \arccos\left(\frac{14}{2.236 \times 6.403}\right) \approx 0.380 \text{ radians} \]
4. Convert to degrees: \[ 0.380 \text{ radians} \times \frac{180°}{\pi} \approx 21.80° \]

Therefore, the angle between \(\vec{a}\) and \(\vec{b}\) is approximately 21.80°.

Example Calculation (3D)

Now let's calculate the angle between two 3D vectors:

\(\vec{a} = (1, 2, 3)\) and \(\vec{b} = (4, 5, 6)\)

1. Calculate the dot product: \[ \vec{a} \cdot \vec{b} = (1 \times 4) + (2 \times 5) + (3 \times 6) = 4 + 10 + 18 = 32 \]
2. Calculate the magnitudes: \[ |\vec{a}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \approx 3.742 \] \[ |\vec{b}| = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{77} \approx 8.775 \]
3. Apply the formula: \[ \theta = \arccos\left(\frac{32}{3.742 \times 8.775}\right) \approx 0.225 \text{ radians} \]
4. Convert to degrees: \[ 0.225 \text{ radians} \times \frac{180°}{\pi} \approx 12.89° \]

Therefore, the angle between \(\vec{a}\) and \(\vec{b}\) in 3D space is approximately 12.89°.

Visual Representation

This scatter plot illustrates the 2D projection of the example vectors, providing a visual representation of the angle between them. Note that for 3D vectors, this is a simplified 2D representation.