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Use the terminology from Section 8.2a. Let $T=\left\{\left[\begin{array}{r}{-1} \\ {0}\end{array}\right],\left[\begin{array}{l}{2} \\ {3}\end{array}\right],\left[\begin{array}{l}{4} \\ {1}\end{array}\right]\right\},$ and let $\mathbf{p}_{1}=\left[\begin{array}{l}{2} \\ {1}\end{array}\right], \mathbf{p}_{2}=\left[\begin{array}{l}{3} \\ {2}\end{array}\right], \mathbf{p}_{3}=\left[\begin{array}{l}{2} \\ {0}\end{array}\right],$ and $\mathbf{p}_{4}=\left[\begin{array}{l}{0} \\ {2}\end{array}\right]$.Find the barycentric coordinates of $\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3},$ and $\mathbf{p}_{4}$ with respect to $T$.b. Use your answers in part (a) to determine whether each of $\mathbf{p}_{1}, \ldots, \mathbf{p}_{4}$ in part (a) is inside, outside, or on the edge of $\mathrm{conv} T,$ a triangular region.

$p _ { 3 }$ and $p _ { 4 }$ are outside conv $T ,$ because they both contain negative coefficient.$p _ { 1 }$ is inside conv $T ,$ becanse barycentric coordinates are all positive.$p _ { 2 }$ is on the edge $\overline { v _ { 1 } v _ { 2 } }$ of conv $T$ because first coordinate is 0 and othersare positive.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

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for part A of this exercise. We're going to use the homogeneous forms off P I on, um, the eyes. So to find the very central coordinates, we have to write the homogeneous form of B I as a linear combination off the homogeneous forms off 31 52 53 on. We can do that by reducing the following augmented matrix. So the first column is gonna be the homogeneous form corresponding off to be one, which is one minus one cereal on, then 123 on, then one for one on. Then on the right hand side, we're going to put the machine is form of B one, which is 1 to 1. Then the machine is from B two. That's it through. So once we reduce this matrix we obtained a fold. First row is gonna be one 00 to 60 0. R. One third. She wrote one half. Well, huh? The second row is going to be 010 1/6 one half when this 1/4 on 9 12, which is also 3/4 on the last row is gonna be serious. 011 half one half 3/4 on minus 1/4. No. Each of these columns represent the virus. Entry coordinates off the P ice Now for part B. We have to say if this points are inside or outside the complex haul off T. So let's take a look at this column which corresponds to be one all off its entries which could respond to the very century coordinates of P one with respectively 1, 15, 53. All of them are positive. They sum up to one eso. This means that this point B one is inside the triangular region. That is the convicts whole off t. Now this column represents speak to on. We can see that there is a entry that is zero on This means well the rest of the entries of this column are positive. So that means that p two is on the edge of this triangular region that represents the convicts Hall of tea. On finally the last two columns which represent Petri MP four, which correspond to be three mp four, we can see that they have negative coordinates. This is enough to say that they are outside off the triangular region, which is the comics hall of tea

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