Status: half-baked
This all is true in any number of dimensions, but let’s say we measure two things for a bunch of people. Perhaps it is height and weight, who knows. A measurement is then a vector \(x = (h, w)\). If we have many measurements we can put them in a matrix \(X = (x_1 \ldots x_n)\).
Suppose we center these data by subtracting the average vector from each individual vector so that \(\tilde x_i = x_i - m\), with \(m = \frac{1}{n} \sum x_i\) being the average. The total variance in these data is \(\frac{1}{n} \sum \lVert \tilde x_i\rVert^2\), the average squared length of the centered vectors.
The variance along a particular vector \(v\) is just the average squared length of your centered vectors if you project them onto \(v\).
For any orthogonal basis \(v_1, v_2\) we can have \(\tilde x = cv_1 + dv_2\) and because of Pythagoras we will have \(\lVert x \rVert^2 = \lVert cv_1 \rVert^2 + \lVert dv_2 \rVert^2\). This means that we can always express the total variance as the sum of variances in orthogonal directions.
