Distance and angle between vectors in multiple dimensions

What does distance and angle between two multidimensional vectors mean? In 3 dimensions, we know what it means because there are at most 3 spatial coordinates, but what does it mean in multiple dimensions?

If we have two multi-dimensional vectors $v,w\in \mathbb{R}^n$ then (assuming that one is not a multiple of the other) together they span a 2-dimensional subspace in $\mathbb{R}^n$. Once you have this, the notion of distance and angle between two vectors falls back into our regular notion of angle and distance in 2-dimensions.
Consider for example $(1,1,1,1), (1,1,0,0)\in\mathbb{R}^4$. This is sort of nice because they span a 2-dimensional subspace that is "almost" one of the coordinate axis subspaces. I recommend that you try writing down a basis for the 2-dimensional subspace they span and changing co-ordinates appropriately, then using this compute the distance and angle. Bear in mind that you should get the following answer.
Alternatively, to calculate the distance and angle between these two vectors we can apply formulae: $$d = \sqrt{(1-1)^2 + (1-1)^2 + (1-0)^2 + (1-0)^2} = \sqrt{2}$$ $$cos\theta = \frac{(1,1,1,1) \cdot (1,1,0,0)}{||(1,1,1,1)|| ||(1,1,0,0)||} = \frac{1}{\sqrt{2}}$$