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. – Hagen von Eitzen Apr 28 '17 at 16:51