This is actually just another way to see the third example (concentration of measure). As you increase the number of dimensions, the contribution of each base v

This is actually just another way to see the third example (concentration of measure). As you increase the number of dimensions, the contribution of each base vector component in the calculation of, say, the cosine angle (i.e. via the scalar product) becomes less important. So in three dimensions you'll have a pretty high angle if one vector component points along a different base vector. But in 10,000 dimensions, the angle will be tiny.