Here are the description about how PCA (Principal component analysis) is calculated (referred from this book). Confused about this statement -- "We can define the direction of this space using a D-dimensional vector u1", I think a vector can represent a direction and length, how could it represent a space? And when talking about PCA, it is projection into a space or projection into a vector (i.e. dot product)?
If $u_1$ is any vector not orthogonal to the line, then we may decree that the projection of $u_1$ onto the line yields the positive direction. So any non-orthogonal vector yields an orientation of the line.