Are 3 Vectors Coplanar? Checking the Triple Product

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The discussion revolves around determining if the vectors (2,1,-1), (1,1,0), and (2,-1,3) are coplanar by calculating the triple product via the determinant of a matrix. Despite an initial belief that the determinant is non-zero, indicating linear independence, it is confirmed to be 6, suggesting the vectors are not coplanar. The confusion arises as coplanarity implies that all vectors lie in the same plane, which is not the case here. The analysis concludes that while two vectors can span a plane, the third vector does not lie within that plane. Ultimately, the vectors are confirmed to be linearly independent and not coplanar.
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so, I'm trying to determine if the vectors
2,1,-1
1,1,0
2,-1,3

are coplanar. i take the triple product, finding the determinant of the matrix. it seems to be non-zero, but the answer key insists these are coplanar. am i wrong, or perhaps the book? any input would be appreciated!
 
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You're right. Did you copy the problem correctly?
 
did you check the vectors out themselves?...? they are coplanar.
 
coplanar means that they all exist on the same plane, right? i just made a vypthon program to draw all of the vectors as they are starting from the origin. it looks to me like the definitely span a parallelpiped, but i could certainly be wrong... now I'm even more confused
 
Well, the determinant is 6, so the three vectors are linearly independent. Since two of the vectors span a plane and the third is not a linear combination of the former two it does not lie in the same plane.
 
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