Properties Of Vector Cross Product

We now note some important properties of the cross product:

(i) If \(\vec a\,\,{\text{and}}\,\,\vec b\) are parallel, their cross product is zero, i.e.

\[\vec a \times \vec b = \vec 0\]

since \(\sin \theta  = 0\). Conversely, if \(\vec a \times \vec b = \vec 0,\) then \(\vec a\,\,{\text{and}}\,\,\vec b\) must be parallel.

(ii) The cross product is not commutative. In fact,

\[\vec b \times \vec a =  - \vec a \times \vec b\]

This is because the direction of \(\vec a \times \vec b\) was defined so that \(\vec a\) ,\(\vec b\) and \(\vec a \times \vec b\) form a right handed system

(iii) The cross product is distributive over vector addition:

\[\begin{align}&\quad\qquad \vec a \times (\vec b + \vec c) = \vec a \times \vec b + \vec a \times \vec c \hfill \\\\&and\quad(\vec a + \vec b) \times \vec c\; = \vec a \times \vec c + \vec b \times \vec c \hfill \\ 
\end{align} \]

(iv)\(\hat i \times \hat i = \hat j \times \hat j = \hat k \times \hat k = \vec 0\)

\(\hat i \times \hat j = \hat k,\;\;\hat j \times \hat k = \hat i,\;\;\hat k \times \hat i = \hat j\)

These relations can be remembered as

Going in the reverse direction, we have

\[\hat j \times \hat i =  - \hat k,\;\;\hat i \times \hat k =  - \hat j,\;\;\hat k \times \hat j =  - \hat i\]

Thus, for two vectors \(\vec a = {a_1}\hat i + {a_2}\hat j + {a_3}\hat k\,\,and\,\,\vec b = {b_1}\hat i + {b_2}\hat j + {b_3}\hat k\)  we have

\[\begin{align}&\vec a \times \vec b \,= ({a_1}\hat i + {a_2}\hat j + {a_3}\hat k) \times ({b_1}\hat i + {b_2}\hat j + {b_3}\hat k) \hfill \\\\&\qquad\;= {a_1}\;{b_2}\hat k - {a_1}{b_3}\hat j - {a_2}{b_1}\hat k + {a_2}{b_3}\hat i + {a_3}{b_1}\hat j - {a_3}{b_2}\hat i \hfill \\\\&\qquad\; = \hat i({a_2}{b_3} - {a_3}{b_2}) + \hat j({a_3}{b_1} - {a_1}{b_3}) + \hat k({a_1}{b_2} - {a_2}{b_1}) \hfill \\ \end{align} \]

This can be written concisely in determinant notation as

\[\vec a \times \vec b = \left| {\begin{align}&{\hat i}&{\hat j}&&{\hat k} \\& {{a_1}}&{{a_2}}&&{{a_3}} \\ & {{b_1}}&{{b_2}}&&{{b_3}} \end{align}} \right|\]

(v) The unit vector(s) \(\hat r\) normal to the plane of \(\vec a\,\,{\text{and}}\,\,\vec b\) can be written as

\[\hat r =  \pm \frac{{\vec a \times \vec b}}{{\left| {\vec a \times \vec b} \right|}}\]