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Choose a set $S$ of four distinct points in $\mathbb{R}^{3}$ such that aff $S$ is the plane $2 x_{1}+x_{2}-3 x_{3}=12 .$ Justify your work.

an affine set$S=\left\{\left(\begin{array}{l}{6} \\ {3} \\ {1}\end{array}\right),\left(\begin{array}{l}{5} \\ {8} \\ {2}\end{array}\right),\left(\begin{array}{l}{2} \\ {8} \\ {0}\end{array}\right),\left(\begin{array}{c}{1} \\ {13} \\ {1}\end{array}\right)\right\}$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

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we solve this question so we because we're told to choose, is sets s or four dick stints points in out for you. So we do these by choosing four points from the blame Sue X one plus X two minus three X Terry. Of course it's solved. So duds at lists we need three off them our notes Kulina. That is why we don't we want want to pick up the street at the local inn is because if the points kulina, the uh oh would be in line I'm not a plane. So sure that those four points known is Karina just into victory. Because if you picked suit as a possibility that they are not soon maybe Kulina sold to pick the three points. So pick three points. That's ah, holding that are not Kalinda. So, for example, we can sets x one equal to X to quart zero on dhe off. We do that off. This implies out of mine a story ex Terry recorded solve on, then extra because of minus four. So with these women, three anos cortina. So we can also said so this is one point second points you can set X one equal to X Treaty Court Zero. This would imply that's when you plug that in. This would imply because X 10 extra is there a Thomas ext Sue Records. It solved. I'm for the foot start points because sets ext Su equal to X tree records zero. This was implied. And after two ex want records to solve and an excellent quarter to six now we need 1/4 points. Um, for the fourth points, we can just pick any points for the fourth for the fourth points. Choose whatever points you want to say. We can choose X one equal to X to the quarter one. You can pick anything you want. So this would be Sue Lost. One minor story. Ex Terry Quarter solve. This implies our ex Terry Quarts of minus three. So this implies that's in total. The points we compete so we can choose as to be sick. Ward. So for the first point, you have x 10 extra 000 minus fall, the second point x 10 extra reserve and we got extra to be solved. Similarly, we odd 600 and then lastly, we have 11 minus

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