# Average of multiple vectors

I have more than two vectors (these are 2D vectors) and I want to calculate the mean vector. What is the correct way to do it? All my vectors share their origins at (0,0).

The mean of a set of vectors is calculated component-wise. In other words, for 2D vectors simply find the mean of the first coordinates and the mean of the second coordinates, and those will be the coordinates of the mean vector.

• Note that this is the same as adding all the vectors using vector addition and then multiplying by $\frac1n$ where $n$ is the number of vectors. In other words, just the same formula as an average of real numbers: $\frac1n(v_1+v_2+\cdots+v_n)$. – Henning Makholm Nov 10 '11 at 20:13

Not sure about this. Let's say we have 3 equally-spaced vectors all of equal length V. They are at 10 deg, 40 deg, 70 deg to the x-axis. Sum the x components and then the y components, then divide each by 3. Then find the resultant. Resultant magnitude is 0.91V, resultant direction is 40 deg. Intuitively, the resultant should be V. If we sum the magnitudes of the vectors and the directions and then divide by 3 we get the correct answer.