Hi!

I understand how we can find the the eigenvectors, what i don't understand is how they help us to map correctly the original points to the PCA world.

for example - in the question i mentioned in the title we are being asked to find the incline of the line we are asked to return with PCA of 1d.

it is clear in this example that a good vector to map each point will be (1,5), and all we need is a line with real value which represent all possible coefficients, a line which can be X axis or Y axis.

but in the general case it won't be so easy to "see" the right vector to map with, and all we get from the PCA is a vector which return for which eigenvectors we should pick. how is this vector helps us to map?