Can the Angle Between Two 7-Dimensional Vectors Be Calculated?

[robertjford80]

Apparently you can calculate the angle between two vectors even when they occupy say 7 dimensions. I have trouble believing that. To me if a vector occupies 7 dimensions than it is incommensurable with another vector occupying 7 dimensions.

 

[slider142]

Two non-collinear vectors span a 2-dimensional plane, no matter how many dimensions they are embedded in. The angle between the vectors is measured in that plane.

 

[HallsofIvy]

Apparently you can calculate the angle between two vectors even when they occupy say 7 dimensions. I have trouble believing that. To me if a vector occupies 7 dimensions than it is incommensurable with another vector occupying 7 dimensions.

First, I don't understand what you mean by a vector "occupying" 7 dimensions. A vector is, pretty much by definition, a one dimensional object, no matter what the dimension of the underlying space. Perhaps that is your misunderstanding. You may be think of the vector as "taking up" the entire space. That is not true in 7 dimensions any more than it is true in 3 dimensions. You are talking about vector s in 7 dimensions, not "occupying" 7 dimensions.