Question

Show the \((a \times b) \cdot b = 0\) for all vectors \({\bf{a}}\) and \({\bf{b}}\) in \({V_3}\).

Short answer

The condition \(({\rm{a}} \times {\rm{b}}) \cdot {\rm{b}} = 0\) is shown for all vectors \({\bf{a}}\) and \({\bf{b}}\) in \({V_3}\).

Step-by-step solution

Formula used

Consider,\({\rm{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \)and\({\rm{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \).

Find the cross product \(({\bf{a}} \times {\bf{b}}) \cdot {\bf{b}}\)

If \({\bf{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle ,{\bf{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \) and \({\bf{c}} = \left\langle {{c_1},{c_2},{c_3}} \right\rangle \), then

\(({\bf{a}} \times {\bf{b}}) \cdot {\bf{c}} = \left| {\begin{array}{*{20}{l}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right|\)

Therefore, if \({\bf{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle ,{\bf{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \), we can write

\(({\bf{a}} \times {\bf{b}}) \cdot {\bf{b}} = \left| {\begin{array}{*{20}{l}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right|\)

Subtract Row 2 from Row 3

\(({\bf{a}} \times {\bf{b}}) \cdot {\bf{b}} = \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\0&0&0\end{array}} \right|\)

If we expand the determinant along the bottom row, we will get

\(\begin{array}{l}({\bf{a}} \times {\bf{b}}) \cdot {\bf{b}} = {( - 1)^{3 + 1}} \cdot 0 \cdot \left| {\begin{array}{*{20}{l}}{{a_2}}&{{a_3}}\\{{b_2}}&{{b_3}}\end{array}} \right| + {( - 1)^{3 + 2}} \cdot 0 \cdot \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_3}}\\{{b_1}}&{{b_3}}\end{array}} \right| + {( - 1)^{3 + 3}} \cdot 0 \cdot \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\\{{b_1}}&{{b_2}}\end{array}} \right|\\({\bf{a}} \times {\bf{b}}) \cdot {\bf{b}} = 0 + 0 + 0\\({\bf{a}} \times {\bf{b}}) \cdot {\bf{b}} = 0\end{array}\)

Hence, the condition \(({\rm{a}} \times {\rm{b}}) \cdot {\rm{b}} = 0\) is shown for all vectors \({\bf{a}}\) and \({\bf{b}}\) in \({V_3}\).