Chapter 13 - Vector Geometry - 13.4 The Cross Product - Exercises - Page 678: 49

Using the formula for the cross product, we obtain ${\bf{v}} \times {\bf{w}} = - {\bf{w}} \times {\bf{v}}$

Let the components of ${\bf{v}}$ and ${\bf{w}}$ be given by ${\bf{v}} = \left( {{v_1},{v_2},{v_3}} \right)$ and ${\bf{w}} = \left( {{w_1},{w_2},{w_3}} \right)$, respectively. Using the formula for the cross product, we have ${\bf{v}} \times {\bf{w}} = \left| {\begin{array}{*{20}{c}} {\bf{i}}&{\bf{j}}&{\bf{k}}\\ {{v_1}}&{{v_2}}&{{v_3}}\\ {{w_1}}&{{w_2}}&{{w_3}} \end{array}} \right|$ ${\bf{v}} \times {\bf{w}} = \left| {\begin{array}{*{20}{c}} {{v_2}}&{{v_3}}\\ {{w_2}}&{{w_3}} \end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}} {{v_1}}&{{v_3}}\\ {{w_1}}&{{w_3}} \end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}} {{v_1}}&{{v_2}}\\ {{w_1}}&{{w_2}} \end{array}} \right|{\bf{k}}$ ${\bf{v}} \times {\bf{w}} = \left( {{v_2}{w_3} - {v_3}{w_2}} \right){\bf{i}} - \left( {{v_1}{w_3} - {v_3}{w_1}} \right){\bf{j}} + \left( {{v_1}{w_2} - {v_2}{w_1}} \right){\bf{k}}$ Similarly, ${\bf{w}} \times {\bf{v}} = \left| {\begin{array}{*{20}{c}} {\bf{i}}&{\bf{j}}&{\bf{k}}\\ {{w_1}}&{{w_2}}&{{w_3}}\\ {{v_1}}&{{v_2}}&{{v_3}} \end{array}} \right|$ ${\bf{w}} \times {\bf{v}} = \left| {\begin{array}{*{20}{c}} {{w_2}}&{{w_3}}\\ {{v_2}}&{{v_3}} \end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}} {{w_1}}&{{w_3}}\\ {{v_1}}&{{v_3}} \end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}} {{w_1}}&{{w_2}}\\ {{v_1}}&{{v_2}} \end{array}} \right|{\bf{k}}$ ${\bf{w}} \times {\bf{v}} = \left( {{w_2}{v_3} - {w_3}{v_2}} \right){\bf{i}} - \left( {{w_1}{v_3} - {w_3}{v_1}} \right){\bf{j}} + \left( {{w_1}{v_2} - {w_2}{v_1}} \right){\bf{k}}$ We may write ${\bf{w}} \times {\bf{v}} = - \left( {{w_3}{v_2} - {w_2}{v_3}} \right){\bf{i}} + \left( {{w_3}{v_1} - {w_1}{v_3}} \right){\bf{j}} - \left( {{w_2}{v_1} - {w_1}{v_2}} \right){\bf{k}}$ ${\bf{w}} \times {\bf{v}} = - \left( {\left( {{v_2}{w_3} - {v_3}{w_2}} \right){\bf{i}} - \left( {{v_1}{w_3} - {v_3}{w_1}} \right){\bf{j}} + \left( {{v_1}{w_2} - {v_2}{w_1}} \right){\bf{k}}} \right)$ But we have from previous result: ${\bf{v}} \times {\bf{w}} = \left( {{v_2}{w_3} - {v_3}{w_2}} \right){\bf{i}} - \left( {{v_1}{w_3} - {v_3}{w_1}} \right){\bf{j}} + \left( {{v_1}{w_2} - {v_2}{w_1}} \right){\bf{k}}$ Therefore, ${\bf{w}} \times {\bf{v}} = - {\bf{v}} \times {\bf{w}}$ ${\ \ }$ or ${\ \ }$ ${\bf{v}} \times {\bf{w}} = - {\bf{w}} \times {\bf{v}}$.