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Consider the points in Exercise 5 in Section $8.1 .$ Which of $\mathbf{p}_{1}, \mathbf{p}_{2},$ and $\mathbf{p}_{3}$ are in conv $S ?$

None of these.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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one way we can use to say if the peace are in the complex, how off s is finding its very central coordinates on that requires to solve the following equation for each eye to solve P I minus v I it goes quiet efficiency, I time speak to minus 31 plus a coefficient C three times speed three minus B one on. Do we have to find the code efficiency one and C three on board? We can do that by reducing the following augmented matrix. So the first column is gonna be be to minus B one the second column, B three minus B one, which is zero minus 60 then p one minus p one p two minus be one on finally be three minus b one. Now if we multiply the 1st and 2nd rose by minus one on if we add yes. So if we multiply the first row by minus one on the first row by minus one we obtained yes, and then we add minus three times the first wrote Third road on We obtained cereal, cereal, civil, my understand in Syria. So from here we can see that in the column corresponding to P two. There's a minus stand at the bottom, which makes the system inconsistent. Eso there is no solution for us. You want to see three when we consider the victory p two on from here? Well, we can conclude that p two does not belong to the convicts. Whole off s Andi. If we ignore this column on we keep producing that ISS. If we subtract the first road the second road we obtained 10 minus one minus four to Oh, sorry, we're going Thio, ignore this second column on, uh 06 minutes 60 and will the last room 00 Um then we divide by six. We obtained 10 minus 12 on 01 minus 10 Syria 000 on from here. Well, we can write that my news be to minus B one minus again, be three minus. B one is equal to P one, minus b one on. If we rewrite this, we have three b one minus speak to minus B three calls p one on. In this linear combination, there are coefficients which are negative on DSO. We conclude that p one is not in the complex. Whole off s on from the last column, which corresponded to P three. We have the following, so we have two times with two minus V one equals p three minus Big one on this can be written us to be to minus 31 It goes to be three on board again. From here we see that the second coefficient is negative. So Petrie is not in the comics haul off s. So in this exercise, none of the peace belonged to the complex Whole off s.

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