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Question Number 44848 by pieroo last updated on 05/Oct/18

$$\left(1\right)\mathrm{If}\overrightarrow{\mathbf{a}}={\mathrm{a}}_{1\mathrm{i}}+{\mathrm{a}}_{2\mathrm{j}}+{\mathrm{a}}_{3\mathrm{k}}\mathrm{and}\overrightarrow{\mathbf{b}}={\mathrm{b}}_{1\mathrm{i}}+{\mathrm{b}}_{2\mathrm{j}}+{\mathrm{b}}_{3\mathrm{k}}\mathrm{show}\mathrm{that}$$$$\mathrm{i}.\overrightarrow{\mathbf{a}}\times \overrightarrow{\mathbf{b}}=\left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ {\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ {\mathrm{b}}_1& {\mathrm{b}}_2& {\mathrm{b}}_3\end{array}\right|$$$$\mathrm{ii}.\overrightarrow{\mathbf{a}}\bullet \overrightarrow{\mathbf{b}}={\mathrm{a}}_1{\mathrm{b}}_1+{\mathrm{a}}_2{\mathrm{b}}_2+{\mathrm{a}}_3{\mathrm{b}}_3$$$$$$$$\left(2\right)\mathrm{If}\overrightarrow{\mathbf{a}}={\mathrm{a}}_{1\mathrm{i}}+{\mathrm{a}}_{2\mathrm{j}}+{\mathrm{a}}_{3\mathrm{k}},\overrightarrow{\mathbf{b}}={\mathrm{b}}_{1\mathrm{i}}+{\mathrm{b}}_{2\mathrm{j}}+{\mathrm{b}}_{3\mathrm{k}}\mathrm{and}$$$$\overrightarrow{\mathbf{c}}={\mathrm{c}}_{1\mathrm{i}}+{\mathrm{c}}_{2\mathrm{j}}+{\mathrm{c}}_{3\mathrm{k}},\mathrm{show}\mathrm{that}$$$$\mathrm{i}.\overrightarrow{\mathbf{a}}\bullet \left(\overrightarrow{\mathbf{b}}\times \overrightarrow{\mathbf{c}}\right)=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ {\mathrm{b}}_1& {\mathrm{b}}_2& {\mathrm{b}}_3\\ {\mathrm{c}}_1& {\mathrm{c}}_2& {\mathrm{c}}_3\end{array}\right|$$$$\mathrm{ii}.\mathbf{a}\bullet \left(\overrightarrow{\mathbf{b}}\times \overrightarrow{\mathrm{c}}\right)=\overrightarrow{\mathbf{b}}(\overrightarrow{\mathbf{a}}\bullet \overrightarrow{\mathbf{c}})-\overrightarrow{\mathbf{c}}(\overrightarrow{\mathbf{a}}\bullet \overrightarrow{\mathbf{b}})$$$$\mathrm{iii}.\left(\overrightarrow{\mathbf{a}}\times \overrightarrow{\mathbf{b}}\right)\times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{b}}(\overrightarrow{\mathbf{a}}\bullet \overrightarrow{\mathbf{c}})-\overrightarrow{\mathbf{a}}(\overrightarrow{\mathbf{b}}\bullet \overrightarrow{\mathbf{c}})$$$$\mathrm{iv}.\overrightarrow{\mathbf{a}}\times \left(\overrightarrow{\mathbf{b}}\times \overrightarrow{\mathbf{c}}\right)+\overrightarrow{\mathbf{b}}\times \left(\overrightarrow{\mathbf{c}}\times \overrightarrow{\mathbf{a}}\right)+\overrightarrow{\mathbf{c}}\times \left(\overrightarrow{\mathbf{a}}\times \overrightarrow{\mathbf{b}}\right)=0$$

Commented by pieroo last updated on 05/Oct/18

$$\mathrm{please}\mathrm{I}\mathrm{need}\mathrm{help}\mathrm{with}\mathrm{the}\mathrm{above}\mathrm{questions}$$

Commented by pieroo last updated on 06/Oct/18

$$\mathrm{still}\mathrm{waiting}\mathrm{for}\mathrm{some}\mathrm{help},\mathrm{please}$$

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