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1-If-a-a-1i-a-2j-a-3k-and-b-b-1i-b-2j-b-3k-show-that-i-a-b-determinant-i-j-k-a-1-a-2-a-3-b-1-b-2-b-3-ii-a-b-a-1-b-1-a-2-b-2-a-3-b-3-2-If-a-




Question Number 44848 by pieroo last updated on 05/Oct/18
(1) If a^→ =a_(1i) +a_(2j) +a_(3k)  and b^→ =b_(1i) +b_(2j) +b_(3k)  show that  i. a^→ ×b^→ = determinant ((i,j,k),(a_1 ,a_2 ,a_3 ),(b_1 ,b_2 ,b_3 ))  ii. a^→ •b^→ =a_1 b_1 +a_2 b_2 +a_3 b_3     (2) If a^→ =a_(1i) +a_(2j) +a_(3k) ,  b^→ =b_(1i) +b_(2j) +b_(3k)  and   c^→ =c_(1i) +c_(2j) +c_(3k) , show that  i. a^→ •(b^→ ×c^→ )= determinant ((a_1 ,a_2 ,a_3 ),(b_1 ,b_2 ,b_3 ),(c_1 ,c_2 ,c_3 ))  ii. a•(b^→ ×c^→ )=b^→ (a^→ •c^→ )−c^→ (a^→ •b^→ )  iii. (a^→ ×b^→ )×c^→ =b^→ (a^→ •c^→ )−a^→ (b^→ •c^→ )  iv. a^→ ×(b^→ ×c^→ )+b^→ ×(c^→ ×a^→ )+c^→ ×(a^→ ×b^→ )=0
$$\left(\mathrm{1}\right)\:\mathrm{If}\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}=\mathrm{a}_{\mathrm{1i}} +\mathrm{a}_{\mathrm{2j}} +\mathrm{a}_{\mathrm{3k}} \:\mathrm{and}\:\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\mathrm{b}_{\mathrm{1i}} +\mathrm{b}_{\mathrm{2j}} +\mathrm{b}_{\mathrm{3k}} \:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{i}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\begin{vmatrix}{\mathrm{i}}&{\mathrm{j}}&{\mathrm{k}}\\{\mathrm{a}_{\mathrm{1}} }&{\mathrm{a}_{\mathrm{2}} }&{\mathrm{a}_{\mathrm{3}} }\\{\mathrm{b}_{\mathrm{1}} }&{\mathrm{b}_{\mathrm{2}} }&{\mathrm{b}_{\mathrm{3}} }\end{vmatrix} \\ $$$$\mathrm{ii}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\mathrm{a}_{\mathrm{1}} \mathrm{b}_{\mathrm{1}} +\mathrm{a}_{\mathrm{2}} \mathrm{b}_{\mathrm{2}} +\mathrm{a}_{\mathrm{3}} \mathrm{b}_{\mathrm{3}} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\:\mathrm{If}\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}=\mathrm{a}_{\mathrm{1i}} +\mathrm{a}_{\mathrm{2j}} +\mathrm{a}_{\mathrm{3k}} ,\:\:\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}=\mathrm{b}_{\mathrm{1i}} +\mathrm{b}_{\mathrm{2j}} +\mathrm{b}_{\mathrm{3k}} \:\mathrm{and}\: \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}=\mathrm{c}_{\mathrm{1i}} +\mathrm{c}_{\mathrm{2j}} +\mathrm{c}_{\mathrm{3k}} ,\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{i}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)=\begin{vmatrix}{\mathrm{a}_{\mathrm{1}} }&{\mathrm{a}_{\mathrm{2}} }&{\mathrm{a}_{\mathrm{3}} }\\{\mathrm{b}_{\mathrm{1}} }&{\mathrm{b}_{\mathrm{2}} }&{\mathrm{b}_{\mathrm{3}} }\\{\mathrm{c}_{\mathrm{1}} }&{\mathrm{c}_{\mathrm{2}} }&{\mathrm{c}_{\mathrm{3}} }\end{vmatrix} \\ $$$$\mathrm{ii}.\:\boldsymbol{\mathrm{a}}\bullet\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\overset{\rightarrow} {\mathrm{c}}\right)=\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)−\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\right) \\ $$$$\mathrm{iii}.\:\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\right)×\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}=\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)−\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\bullet\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right) \\ $$$$\mathrm{iv}.\:\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\left(\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}\right)+\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}×\left(\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}\right)+\overset{\rightarrow} {\boldsymbol{\mathrm{c}}}×\left(\overset{\rightarrow} {\boldsymbol{\mathrm{a}}}×\overset{\rightarrow} {\boldsymbol{\mathrm{b}}}\right)=\mathrm{0} \\ $$
Commented by pieroo last updated on 05/Oct/18
please I need help with the above questions
$$\mathrm{please}\:\mathrm{I}\:\mathrm{need}\:\mathrm{help}\:\mathrm{with}\:\mathrm{the}\:\mathrm{above}\:\mathrm{questions} \\ $$
Commented by pieroo last updated on 06/Oct/18
still waiting for some help, please
$$\mathrm{still}\:\mathrm{waiting}\:\mathrm{for}\:\mathrm{some}\:\mathrm{help},\:\mathrm{please} \\ $$

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