Question

For any three vectors a.b.c. write the value of the following: a cross (b+c) + b cross (c+a) + c cross (a+b).

Answer 1

$$\begin{gathered} \overrightarrow{\mathrm{a}}\times \left(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\right)+\overrightarrow{\mathrm{b}}\times \left(\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}\right)+\overrightarrow{\mathrm{c}}\times \left(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right) \\ =\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{b}} \\ =\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{b}} \\ =\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}} \\ =\overrightarrow{0} \end{gathered}$$