Question

Find the cross product between \({\rm{a}}\) and \({\rm{b}}\) and \({\rm{b}}\) and \({\rm{a}}\).

Short answer

\(a \times b\)is\( - 7i + 10j + 8k\).

\(b \times a\) is \(7i - 10j - 8k\).

Step-by-step solution

Formula used

Consider the general expression to find the cross product of\({\bf{a}}\)and\({\bf{b}}\).

\({\rm{a}} \times {\rm{b}} = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right| \ldots \ldots \ldots (1)\)

Consider the general expression to find the cross product of\({\bf{b}}\)and\({\bf{a}}\).

\({\rm{b}} \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right| \ldots \ldots \ldots (2)\)

Find the \(a \times b\)

As given, \(a = \langle 2, - 1,3\rangle \) and \({\rm{b}} = \langle 4,2,1\rangle \)

Substitute 2 for \({a_1}, - 1\) for \({a_2},3\) for \({a_3},4\) for \({b_1},2\) for \({b_2}\) and 1 for \({b_3}\) in equation (1),

\(\begin{array}{l}a \times b = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\2&{ - 1}&3\\4&2&1\end{array}} \right|\\a \times b = \left| {\begin{array}{*{20}{c}}{ - 1}&3\\2&1\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}2&3\\4&1\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{c}}2&{ - 1}\\4&2\end{array}} \right|{\rm{k}}\\a \times b = ( - 1 - 6){\rm{i}} - (2 - 12){\rm{j}} + (4 + 4){\rm{k}}\\a \times b = - 7{\rm{i}} + 10{\rm{j}} + 8{\rm{k}}\end{array}\)

Thus, the \(a \times b\) is \( - 7i + 10j + 8k\).

Find the \(b \times a\)

Substitute 2 for \({a_1}, - 1\) for \({a_2},3\) for \({a_3},4\) for \({b_1},2\) for \({b_2}\) and 1 for \({b_3}\) in equation (2),

\(\begin{array}{l}{\rm{b}} \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\4&2&1\\2&{ - 1}&3\end{array}} \right|\\{\rm{b}} \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}2&1\\{ - 1}&3\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}4&1\\2&3\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{c}}4&2\\2&{ - 1}\end{array}} \right|{\rm{k}}\\{\rm{b}} \times {\rm{a}} = (6 + 1){\rm{i}} - (12 - 2){\rm{j}} + ( - 4 - 4){\rm{k}}\\{\rm{b}} \times {\rm{a}} = 7{\rm{i}} - 10{\rm{j}} - 8{\rm{k}}\end{array}\)

Thus, the \(b \times a\) is \(7i - 10j - 8k\).