Question Number 42199 by rahul 19 last updated on 20/Aug/18
Answered by tanmay.chaudhury50@gmail.com last updated on 20/Aug/18
![a){(a+b)×(b+c)}.(c+a) (a×b+a×c+b×b+b×c).(c+a) (a×b).c+(b×c).a=2v [abc]=[bca]=[cab]=v b){(a−b)×(b+c)}.(c+a) (a×b+a×c−b×c).(c+a) =(a×b).c−(b×c).a=0 c){(a+b)×(b−c)}.(c+a) =(a×b−a×c−b×c).(c+a) =(a×b).c−(b×c).a=0 d){(a+b)×(b+c)}.(c−a) =(a×b+a×c+b×c).(c−a) =(a×b).c−(b×c).a=0 so option b,c and d volume zero so coplaner](https://www.tinkutara.com/question/Q42206.png)
$$\left.{a}\right)\left\{\left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)×\left(\boldsymbol{{b}}+\boldsymbol{{c}}\right)\right\}.\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right) \\ $$$$\left(\boldsymbol{{a}}×\boldsymbol{{b}}+\boldsymbol{{a}}×\boldsymbol{{c}}+\boldsymbol{{b}}×\boldsymbol{{b}}+\boldsymbol{{b}}×\boldsymbol{{c}}\right).\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right) \\ $$$$\left(\boldsymbol{{a}}×\boldsymbol{{b}}\right).\boldsymbol{{c}}+\left(\boldsymbol{{b}}×\boldsymbol{{c}}\right).\boldsymbol{{a}}=\mathrm{2}{v} \\ $$$$\left[{abc}\right]=\left[{bca}\right]=\left[{cab}\right]={v} \\ $$$$\left.{b}\right)\left\{\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)×\left(\boldsymbol{{b}}+\boldsymbol{{c}}\right)\right\}.\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right) \\ $$$$\left(\boldsymbol{{a}}×\boldsymbol{{b}}+\boldsymbol{{a}}×\boldsymbol{{c}}−\boldsymbol{{b}}×\boldsymbol{{c}}\right).\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right) \\ $$$$=\left(\boldsymbol{{a}}×\boldsymbol{{b}}\right).\boldsymbol{{c}}−\left(\boldsymbol{{b}}×\boldsymbol{{c}}\right).\boldsymbol{{a}}=\mathrm{0} \\ $$$$\left.\boldsymbol{{c}}\right)\left\{\left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)×\left(\boldsymbol{{b}}−\boldsymbol{{c}}\right)\right\}.\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right) \\ $$$$=\left(\boldsymbol{{a}}×\boldsymbol{{b}}−\boldsymbol{{a}}×\boldsymbol{{c}}−\boldsymbol{{b}}×\boldsymbol{{c}}\right).\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right) \\ $$$$=\left(\boldsymbol{{a}}×\boldsymbol{{b}}\right).\boldsymbol{{c}}−\left(\boldsymbol{{b}}×\boldsymbol{{c}}\right).\boldsymbol{{a}}=\mathrm{0} \\ $$$$\left.\boldsymbol{{d}}\right)\left\{\left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)×\left(\boldsymbol{{b}}+\boldsymbol{{c}}\right)\right\}.\left(\boldsymbol{{c}}−\boldsymbol{{a}}\right) \\ $$$$=\left(\boldsymbol{{a}}×\boldsymbol{{b}}+\boldsymbol{{a}}×\boldsymbol{{c}}+\boldsymbol{{b}}×\boldsymbol{{c}}\right).\left(\boldsymbol{{c}}−\boldsymbol{{a}}\right) \\ $$$$=\left(\boldsymbol{{a}}×\boldsymbol{{b}}\right).\boldsymbol{{c}}−\left(\boldsymbol{{b}}×\boldsymbol{{c}}\right).\boldsymbol{{a}}=\mathrm{0} \\ $$$$\boldsymbol{{s}}{o}\:{option}\:{b},{c}\:{and}\:{d}\:{volume}\:{zero}\:{so}\:{coplaner} \\ $$$$ \\ $$
Commented by rahul 19 last updated on 20/Aug/18
thanks sir ��