The dot product (scalar product) of two vectors is a . b = a_{1}$$b_{1} + a_{2}$$b_{2} + a_{3}$$b_{3} = |a||b|cos(theta). It produces a scalar and is commutative and distributive. The fundamental results are: i.i = j.j = k.k = 1, and i.j = j.k = k.i = 0. The angle between vectors is theta = c$os^{-1}$(a.b/(|a||b|)). Two non-zero vectors are perpendicular iff a.b = 0. The scalar projection of b onto a is a.b/|a|, and the vector projection is (a.b/|a|^{2})a. Key identity: a.a = |a|^{2}. For the expansion identity: |a + b|^{2} = |a|^{2} + 2a.b + |b|^{2} and (a+b).(a-b) = |a|^{2} - |b|^{2}. The dot product is used extensively in JEE for finding angles, checking perpendicularity, computing projections, calculating work done by forces, and as an intermediate step in triple product calculations.