InterviewSolution
 Saved Bookmarks
    				| 1. | 
                                    Three non - zero vectors bar(a),bar(b),bar(c) are complanar if and only if there exist scalars x,y,z, not all zero simultaneously such that xbar(a)+ybar(b)+zbar(c)=bar(0). | 
                            
| 
                                   
Answer» Solution :Let `bar(a),bar(b),bar(c)` be coplanar vectors. Then any ONE of them, say `bar(a)`, will be the linear combination of `bar(b)andbar(c)`.  `:.` there EXIST scalars `alphaandbeta` such that `bar(a)=alphabar(b)+betabar(c)` `:.(-1)bar(a)+alphabar(b)+betabar(c)=bar(0),i.e.,xbar(a)+ybar(b)+zbar(c)=bar(0)` where `X=-1,y=ALPHA,z=beta` which are not all zero simultaneously. Conversely : Let there exist scalars x,y,z not all zero such that `xbar(a)+ybar(b)+zbar(c)=bar(0)`. . . (1) Let `x!=0`, then divide (1) by s, we GET `i.e.,bar(a)+((y)/(x))bar(b)+((z)/(x))bar(c)=bar(0)` `:.bar(a)=(-(y)/(x))bar(b)+(-(z)/(x))bar(c)` `i.e.,bar(a)=alphabar(b)+betabar(c)`, where `alpha=(-y)/(x)andbeta=(-z)/(x)` are scalars. `:.bar(a)` is the linear combination of `bar(b)andbar(c)`. Hence, `bar(a),bar(b),bar(c)` are coplanar.  | 
                            |