If ` vec a ,\ vec b ,\ vec c` are three non coplanar vectors such that ` vec ddot vec a= vec ddot vec b= vec ddot vec c=0,` then show that `d` is the null vector.

Explanation

Given that,
` vec a ,\ vec b ,\ vec c`
are three non coplanar vectors
also,
`vec d * vec a= vec d * vec b= vec d * vec c =0`
`"CASE-1:"`
` vec d * vec a=0 `
`=> vec d =0` or `vec d bot vec a`
`"CASE-2:"`
` vec d * vec b=0 `
`=> vec d =0` or `vec d bot vec b`
`"CASE-3:"`
` vec d * vec c=0 `
`=> vec d =0` or `vec d bot vec c`
But vec d can not be perpendicular to all `vec a, vec b and vec c` at a time, as ` vec a ,\ vec b ,\ vec c`
are three non coplanar vectors
so,` vec d =0`
Hence vec d is a null vector.