The dot product (scalar product) is defined as a . b = a_{1}$$b_{1} + a_{2}$$b_{2} + a_{3}$$b_{3} = |a||b|cos(theta), where theta is the angle between the vectors (0 <= theta <= pi). Properties: (1) Commutative: a.b = b.a. (2) Distributive: a.(b+c) = a.b + a.c. (3) a.a = |a|^{2}. (4) i.i = j.j = k.k = 1 and i.j = j.k = k.i = 0. The angle between two vectors is theta = c$os^{-1}$(a.b / (|a||b|)). Two non-zero vectors are perpendicular iff a.b = 0.