# Angle between random vectors with uniformly distributed coordinates

I have been experimenting with random vectors in $$d$$ dimensions, where each coordinate is independently generated from a uniform distribution between 0 and 1. This is

$$X = (x_1, x_2, ..., x_d) \\ Y = (y_1, y_2, ..., y_d)$$

where $$x_i, y_j \sim U(0, 1)$$.

Upon numerically computing the cosine similarity between two such vectors, I consistently observe an approximation of

$$\text{cos sim}(X, Y) = \frac{X \cdot Y}{ \left \| X\right \| \left \| Y \right \|} \approx 0.75$$

I am intrigued by this numerical result and am seeking a mathematical explanation or derivation for this specific value. Moreover, I've noticed that the result doesn't change too much as $$d$$ changes. It's worth noting that the vectors are not uniformly distributed on the $$d$$-dimensional sphere (if this was the case I know the angle between the vectors goes to $$0$$ as $$d\to \infty$$).

Can someone shed light on why this particular value emerges and provide insights into the underlying mathematical reasoning?

For large $$d$$, this value will be close to the value you get if you substitute the expected value for each factor.

The expected value of $$X\cdot Y$$ and of $$\|X\|^2$$ is in each case $$n$$ times the expected value of each of the $$n$$ summands. For $$X\cdot Y$$, that’s

$$\int_0^1\mathrm dx\int_0^1\mathrm dy xy=\frac12\cdot\frac12=\frac14\;.$$

For $$\|X\|^2$$, it’s

$$\int_0^1\mathrm dxx^2=\frac13\;.$$

Thus, for large $$d$$ the quotient goes to

$$\frac{\frac14}{\frac13}=\frac34\;.$$

• Are you using something like law of large numbers to actually pass to the limit here? Jan 30 at 13:43