How to Calculate the Angle Between Two Vectors

The angle between two vectors is defined using dot products and trigonometry. There are various ways to calculate the angle, but the simplest way to find this angle when working with two-dimensional vectors is to take the arccosine of the dot product of the two vectors.

Things You'll Need

• Two defined vectors
• Calculator (optional)

Instructions

Calculate the Angle

1. For vectors A and B, calculate the dot product of two vectors using the formula Ax*Bx + Ay*By. For example, if you had vector A = (4,2) and vector B = (5, 7) the dot product would equal 4*5 + 2*7.

2. You now need to take the arc cosine (acos) of the dot product. To start, square the dot product and then subtract one from the square. Take the square root of this number.

3. Add the number you now have to the original dot product. Take the natural log of this number and that is the angle.

Tips & Warnings

• If you have use of a graphing calculate it will calculate the arc cosine on its own. Simply calculate the dot product and then plug it in to your calculator.

• If you are working with a three dimensional vector you will also need to calculate the axis.

References

• Euclidean Space: Angle Between Vectors