Question

Show the equation \(0 \times {\rm{a}} = 0 = {\rm{a}} \times 0\) for any vector \({\rm{a}}\) in \({V_3}\).

Short answer

The equation \(0 \times {\rm{a}} = 0 = {\rm{a}} \times 0\) is shown for any vector \({\rm{a}}\) in \({V_3}\).

Step-by-step solution

Formula used

Consider,\({\rm{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \).

Find the cross product between \(0\) and \({\rm{a}}\)

\(\begin{array}{l}0 \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\0&0&0\\{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right|\\0 \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}0&0\\{{a_2}}&{{a_3}}\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}0&0\\{{a_1}}&{{a_3}}\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{c}}0&0\\{{a_1}}&{{a_2}}\end{array}} \right|{\rm{k}}\\0 \times {\rm{a}} = (0 - 0){\rm{i}} - (0 - 0){\rm{j}} + (0 - 0){\rm{k}}\\0 \times {\rm{a}} = 0\end{array}\)

Find the cross product between \({\rm{a}}\) and \(0\)

Find the cross product between a and 0

\(\begin{array}{l}a \times 0 = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{a_1}}&{{a_2}}&{{a_3}}\\0&0&0\end{array}} \right|\\a \times 0 = \left| {\begin{array}{*{20}{c}}{{a_2}}&{{a_3}}\\0&0\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_3}}\\0&0\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\\0&0\end{array}} \right|{\rm{k}}\\a \times 0 = (0 - 0){\rm{i}} - (0 - 0){\rm{j}} + (0 - 0){\rm{k}}\\a \times 0 = 0\end{array}\)

Here, it concludes that\(0 \times {\rm{a}} = 0 = {\rm{a}} \times 0\).

Therefore, the equation \(0 \times {\rm{a}} = 0 = {\rm{a}} \times 0\) is shown for any vector \(a\) in \({V_3}\).