Figure 2

The projection of y(y _p ) onto the plane spanned by x and z comprises the appropriate proportions of the vectors x and z where the proportion of vector x is b _X and the proportion of z is b _Z . These proportions are derived by means of the parallelogram rule: y _p is projected onto x parallel to the direction of z to obtain the proportion b _X of x. Likewise, y _p is projected onto z parallel to the direction of x to obtain the proportion b _Z of z. The P-value for the partial regression coefficient of X(b _X ), when Y is regressed on X whilst also adjusting for Z, is derived within vector geometry from the projection of y and x onto the subspace perpendicular to z (V _⊥z ).