Dot Product and Cross Product

Dot Product of Two Vectors

In 3-space, the dot product of two vectors is a scalar. The dot product of vectors \boldsymbol{a} = < a_1,a_2,a_3 > and \boldsymbol{b} = < b_1,b_2,b_3 > is

(1)   \begin{equation*} \boldsymbol{a} . \boldsymbol{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 .\end{equation*}


There is also an alternative form of

(2)   \begin{equation*} \boldsymbol{a} . \boldsymbol{b} = || \boldsymbol{a} || ~ || \boldsymbol{b} || \cos \theta , \end{equation*}


where \theta is the angle between the two vectors.

Cross Product of Two Vectors

In 3-space, the cross product of two vectors is a vector. The cross product of vectors \boldsymbol{a} = < a_1,a_2,a_3 > and \boldsymbol{b} = < b_1,b_2,b_3 > is

(3)   \begin{equation*} \boldsymbol{a} \times \boldsymbol{b} = ( a_2 b_3 - a_3 b_2) \boldsymbol{i} - (a_1 b_3 - a_3 b_1) \boldsymbol{j} + (a_1 b_2 - a_2 b_1 ) \boldsymbol{k} . \end{equation*}


There is also an alternative form of

(4)   \begin{equation*} \boldsymbol{a} \times \boldsymbol{b} = || \boldsymbol{a} || ~ || \boldsymbol{b} || \sin \theta , \end{equation*}


where \theta is the angle between the two vectors.

Orthogonal Vectors

Two non-zero vectors \boldsymbol{a} and \boldsymbol{b} are orthogonal, if and only if, ~ \boldsymbol{a} . \boldsymbol{b} = 0.

Parallel Vectors

Two non-zero vectors \boldsymbol{a} and \boldsymbol{b} are parallel, if and only if, ~ \boldsymbol{a} \times \boldsymbol{b} = 0.

A 12-Minute Lecture on Vector Products