Figure 1

(a) The correlation between variables Y and X (ρ _XY ) is the cosine of θ _xy , the angle between vectors x and y; the projection of y on x (denoted y _p ) has the length ||y||·cos(θ _xy ). Vector y _p lies in the same direction as vector x and may therefore be expressed as a multiple of x: y _p = b _X x, where b _X = (||y||/||x||)cos(θ _xy ) – the simple regression coefficient for X when Y is regressed on X. (b) If θ _xy = 90° (i.e. π/2 radians), then x and y are orthogonal (denoted x ⊥ y), the correlation between X and Y is zero: ρ _XY = cos(90°) = cos(π/2) = 0 and the regression coefficient for Y regressed on X is also zero.