Not really, for example, in physics, lines in 4D are just as meaningful as they are in 3D, more even (they are called geodesics). So are the angles between them. The real problem is that we just don't have good intuitions of higher dimensions in general.
that these are perpendicular to each other, which I will easily call ninety degrees, and that two such collinear vectors are at zero degrees.
But I somehow wouldn't go from that intuition into specific cosines. Like "Oh, look, if I divide out the lengths from the dot product, I'm getting 0.5! Why that's the cosine of 60 degrees!"
