Question Number 44848 by pieroo last updated on 05/Oct/18
$$\left(\mathrm{1}\right)\:\mathrm{If}\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}=\mathrm{a}_{\mathrm{1i}} +\mathrm{a}_{\mathrm{2j}} +\mathrm{a}_{\mathrm{3k}} \:\mathrm{and}\:\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\mathrm{b}_{\mathrm{1i}} +\mathrm{b}_{\mathrm{2j}} +\mathrm{b}_{\mathrm{3k}} \:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{i}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\begin{vmatrix}{\mathrm{i}}&{\mathrm{j}}&{\mathrm{k}}\\{\mathrm{a}_{\mathrm{1}} }&{\mathrm{a}_{\mathrm{2}} }&{\mathrm{a}_{\mathrm{3}} }\\{\mathrm{b}_{\mathrm{1}} }&{\mathrm{b}_{\mathrm{2}} }&{\mathrm{b}_{\mathrm{3}} }\end{vmatrix} \\ $$$$\mathrm{ii}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\mathrm{a}_{\mathrm{1}} \mathrm{b}_{\mathrm{1}} +\mathrm{a}_{\mathrm{2}} \mathrm{b}_{\mathrm{2}} +\mathrm{a}_{\mathrm{3}} \mathrm{b}_{\mathrm{3}} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\:\mathrm{If}\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}=\mathrm{a}_{\mathrm{1i}} +\mathrm{a}_{\mathrm{2j}} +\mathrm{a}_{\mathrm{3k}} ,\:\:\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\mathrm{b}_{\mathrm{1i}} +\mathrm{b}_{\mathrm{2j}} +\mathrm{b}_{\mathrm{3k}} \:\mathrm{and}\: \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}=\mathrm{c}_{\mathrm{1i}} +\mathrm{c}_{\mathrm{2j}} +\mathrm{c}_{\mathrm{3k}} ,\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{i}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)=\begin{vmatrix}{\mathrm{a}_{\mathrm{1}} }&{\mathrm{a}_{\mathrm{2}} }&{\mathrm{a}_{\mathrm{3}} }\\{\mathrm{b}_{\mathrm{1}} }&{\mathrm{b}_{\mathrm{2}} }&{\mathrm{b}_{\mathrm{3}} }\\{\mathrm{c}_{\mathrm{1}} }&{\mathrm{c}_{\mathrm{2}} }&{\mathrm{c}_{\mathrm{3}} }\end{vmatrix} \\ $$$$\mathrm{ii}.\:\boldsymbol{\mathrm{a}}\bullet\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\overset{\rightarrow} {\mathrm{c}}\right)=\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)−\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\right) \\ $$$$\mathrm{iii}.\:\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\right)×\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}=\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)−\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right) \\ $$$$\mathrm{iv}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)+\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\left(\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\right)+\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}×\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\right)=\mathrm{0} \\ $$
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
$$\mathrm{still}\:\mathrm{waiting}\:\mathrm{for}\:\mathrm{some}\:\mathrm{help},\:\mathrm{please} \\ $$