# What is the mean of 3 independent vectors?

Say I want to measure a 2D displacement 3 times and find an average. I measure the X and Y co-ordinates relative to a datum point find a displacement vector consisting of the magnitude and direction (direction relative to a fixed horizontal line). Since the 3 vectors are measured separately, they are independent. How do we find the mean value? This might equally apply to other vector measurements, e.g., forces.

If we separate each vector into X and Y components we can sum each component and divide by 3 and then find the resultant using Pythagoras. This is the answer previously given to a similar question posed by Ali. However, look at this:-

Assume each vector magnitude is the same (V) and that the angles to the horizontal (x direction) are 10 deg, 40 deg, 70 deg. Then SumY/3 = 0.585, SumX/3 = 0.698, and the resultant becomes 0.91V at an angle 40 deg. Intuitively, the mean should be V. If we simply take the arithmetic mean of the vector magnitudes and of the directions we get mean = V, angle = 40 deg. Is the latter correct?

• Intuition is wrong. Apr 28, 2017 at 16:51

## 2 Answers

There is no magic answer. Another approach is to express each vector in magnitude and angle and average the magnitudes and angles. Averaging angles has problems if they are widely scattered but if the range is small it is easy to make sense of it. The point of averaging is to get a "most likely" value from noisy input data. The best way to do that depends on the characteristics of your noise. In your example the average would be V at 40 degrees. Is that what you want?

The answer depends on what you think the average value should mean and what you intend to do with it. We don't add magnitudes when we do vector addition, so there's no a priori reason we should always add them for averaging--but in some applications, where the vector magnitude has a special significance, it might make sense to do that. ("Displacement" doesn't sound like such an application, except possibly if it is displacement from the center of a central force field.)