Imagining a Linear Regression model with more than 3 dimensions
Humans live in a 3 dimensional world. Consequently it can be hard for us to visualise anything beyond 3 dimensions. For 4 dimensions, we can perhaps think about time as the next dimension, so we might think about how a 3 dimensional image changes over time. Even that is quite tricky, so for 5 dimensions and above it is basically impossible.
Rather than trying to visualise a model in terms of geometry, it may be better to understand how models work by using linear algebra (ie matrix algebra)