Prove that the following vectors are coplanar: `2 hat i- hat j+ hat k ,\ hat i-3 hat j-5 hat k\ a n d\ 3 hat i-4 hat j-4 hat k`

Explanation

We are given the vectors


`vec A=2 hat i-hat j+ hat k`


`vec B= hat i-3 hat j-5 hat k`


`vec C=3 hat i-4 hat j-4 hat k`


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

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


`|[2,-1,1],[1,-3,-5],[3,-4,-4]|`


`=2[(-3)(-4)-(-4)(-5)]-(-1)[(1)(-4)-(3)(-5)]+(1)[(1)(-4)-(-3)(3)]`


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