Angle Between Two Vectors Calculator

How to calculate the angle between vectors?

The angle between two vectors is derived from the geometric definition of the dot product, which states that the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.

θ = arccos((A · B) / (|A| |B|))

Using the Vector Angle calculator: an example

Suppose we have Vector A = (3, 4) and Vector B = (4, -3). To find the angle between them, follow these steps:

Step-by-step calculation:

1. Calculate the dot product: (3 * 4) + (4 * -3) = 12 - 12 = 0
2. Calculate magnitudes: |A| = √(3² + 4²) = 5, |B| = √(4² + (-3)²) = 5
3. Apply the formula: θ = arccos(0 / (5 * 5)) = arccos(0)
4. Determine the angle: 90°

Frequently Asked Questions

• What if the dot product is zero?
  If the dot product is zero, the vectors are orthogonal, which means they are perpendicular to each other.
• Can the angle be greater than 180 degrees?
  No, by definition, the angle between two vectors is always the smallest angle between them, ranging from 0 to 180 degrees.
• What does a negative dot product mean?
  A negative dot product indicates that the angle between the vectors is obtuse, meaning it is greater than 90 degrees.