Prove that a necessary and sufficient condition for three vectors ` vec a , vec b` and ` vec c` to be coplanar is that there exist scalars `l , m , n` not all zero simultaneously such that `l vec a+m vec b+n vec c= vec0dot`

Explanation

We are given that


`l vec a+m vec b+n vec c= vec0`

be coplanar then,


`vec c=-1/n(l vec a+m vec b)`


`rArr vec c=pvec a+q vec b`


where, `p=-l/n` and `q=-m/n`


this shows that the vectors `vec a, vec b \ and \ vec c` are coplanar.


For sufficient condition, if `vec a, vec b, \ and \ vec c` are coplanar then there is a condition that


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