If ` vec a ,\ vec b ,\ vec c` are non coplanar vectors, prove that the following vectors are non coplanar:`\ vec a+2 vec b+3 vec c ,\ 2 vec a+ vec b+3 vec c\ a n d\ vec a+ vec b+ vec c`

Explanation

We are given the vectors


`vec A=vec a+2 vec b+3 vec c`


`vec B= 2 vec a+ vec b+3 vec c`


`vec C= vec a+ vec b+ vec c`


Also it is given that `vec a, vec b, vec c` are non-coplanar


Now to show non-coplanar of the vectors we must satisfy the condition

`[ vec(A) quad vec(B) quad vec(C)] != 0`


`rArr |[1,2,3],[2,1,3],[1,1,1]|`


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