Question: For which real constants a and b does the vectors [(1,a,0),(a,1,1),(-b,-1,-1)] lie in the same plane? So know that the lines are in the same plane when the determinant is 0 so I've calulated it like this: \[A=\left[\begin{matrix}1 & a & 0 \\ a & 1 & 1 \\-b &-1&-1 \end{matrix}\right]\] \[\det(A)=1\left[\begin{matrix}1 & 1 \\ -1 & -1\end{matrix}\right]-a \left[\begin{matrix}a & 1\\ -b & -1\end{matrix}\right]+0= -a(-a+b)=a( a-b)\] \[a(a-b)=0 \] \[a=0, b=a\] It's right but the student solution guide solve it \[\left[\begin{matrix}1 & a & -b \\ a & 1&-1 \\ 0&1&-1\end{matrix}\right]\]

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Question: For which real constants a and b does the vectors [(1,a,0),(a,1,1),(-b,-1,-1)] lie in the same plane? So know that the lines are in the same plane when the determinant is 0 so I've calulated it like this: \[A=\left[\begin{matrix}1 & a & 0 \\ a & 1 & 1 \\-b &-1&-1 \end{matrix}\right]\] \[\det(A)=1\left[\begin{matrix}1 & 1 \\ -1 & -1\end{matrix}\right]-a \left[\begin{matrix}a & 1\\ -b & -1\end{matrix}\right]+0= -a(-a+b)=a( a-b)\] \[a(a-b)=0 \] \[a=0, b=a\] It's right but the student solution guide solve it \[\left[\begin{matrix}1 & a & -b \\ a & 1&-1 \\ 0&1&-1\end{matrix}\right]\]

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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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they are using that one as a determinate, is my solution equally right or should I spend some time learn there way?
equally right. it's a property of the determinant that transposing it will not change the value of the determinant
Thank you! :)

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you're welcome :)

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