Computing scalar/dot product between principal component and data I am very new to R and statistics so this may be a simple question. I have a matrix (1000,756) containing 1000 years of winter sea-level pressure data (SLP) at 756 locations in the North Atlantic. I need to identify an oscillation in SLP anomalies (i.e. the difference between high and low regions in the North and South), called the North Atlantic Oscillation.
I have done a principal component analysis of the data using princomp. According to the literature I need to use the leading PC and 

... project the time series of SLP anomaly ﬁelds on to this pattern (i.e. compute the scalar or dot product between field and pattern).

Can anyone help me with how to do this?
 A: The first principal axis (some people would refer to it as "principal component", but I advocate calling it "principal axis") is a vector of length $756$ (number of your "locations"), so $756$ numbers $w_i$. To project your data onto the first principal axis you need to take each column of the data matrix (i.e. $1000$ years at one location), multiply it by the corresponding number $w_i$ (the whole column is multiplied by the same number), and add the $756$ resulting $1000$-long columns together. You will get one column of length $1000$, and this is your North Atlantic Oscillation. This projection is also what I would call "principal component".
Important thing to realize is that principal axis is a vector in location space, i.e. has as many coordinates as there are locations. Projecting onto this axis means reducing all your $756$ locations to one "composite location", which is simply a linear combination (i.e. "weighted sum") of all individual locations.

See my answer here about this terminological distinction: What exactly is called "principal component" in PCA?
