How to Find the Angles Between Two Vectors?

To find the angle between two vectors, say vector a and vector b, you can use the formula derived from the dot product:

cos(θ) = (a • b) / (||a|| ||b||)

Where:

• θ = angle between the vectors
• a • b = dot product of vectors a and b
• ||a|| = magnitude (length) of vector a
• ||b|| = magnitude (length) of vector b

Here’s how you can calculate it step-by-step:

1. First, calculate the dot product of vectors a and b. If a = (a1, a2, a3) and b = (b1, b2, b3), then:

a • b = a1 * b1 + a2 * b2 + a3 * b3

2. Next, calculate the magnitude of each vector:

||a|| = √(a1² + a2² + a3²)

||b|| = √(b1² + b2² + b3²)

3. Now substitute these values into the cosine formula:

cos(θ) = (a • b) / (||a|| ||b||)

4. Finally, take the inverse cosine (arccos) to find the angle θ:

θ = cos^-1((a • b) / (||a|| ||b||))

This will give you the angle between the two vectors in radians. Convert it to degrees if necessary by using:

degrees = radians * (180/π)