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In Exercises $21-24, \mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ are noncollinear points in $\mathbb{R}^{2}$ and $\mathbf{p}$ is any other point in $\mathbb{R}^{2} .$ Let $\Delta \mathbf{a} \mathbf{b} \mathbf{c}$ denote the closed triangularregion determined by $\mathbf{a}, \mathbf{b},$ and $\mathbf{c},$ and let $\Delta \mathbf{p} \mathbf{b} \mathbf{c}$ be the region determined by $\mathbf{p}, \mathbf{b},$ and $\mathbf{c} .$ For convenience, assume that $\mathbf{a}, \mathbf{b},$ and c are arranged so that det $[\tilde{\mathbf{a}} \quad \tilde{\mathbf{b}} \quad \tilde{\mathbf{c}}]$ is positive, where $\tilde{\mathbf{a}}, \tilde{\mathbf{b}},$ and $\tilde{\mathbf{c}}$ are the standard homogeneous forms for the points.

Show that the area of $\Delta \mathrm{abc}$ is det $\left[\begin{array}{lll}{\tilde{\mathbf{a}}} & {\mathbf{b}} & {\tilde{\mathbf{c}}}\end{array}\right] / 2 .[\text { Hint: Con- }$ sult Sections 3.2 and $3.3,$ including the Exercises.]

See explanation.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 2

Affine Independence

Vectors

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So first. Now we can translate the Strangler region to the origin by subtracting a on the area of the triangle region is gonna be one half of the area of the parallelogram determined by B minus A and C minus A that is one half off the determinant off the matrix whose columns are B minus a on C minus e. There's determinant. Remember, that represents the area of the parallelogram on the other side. If we want to compute the determinant of the matrix whose columns are the homogeneous forms off a V and C, then we can write this as so remember that the on my Penis forms are result by adding, uh, coordinate whose value is one. So we have this matrix on Don't remember also that the if we at a multiple of one column to another, then the determinant doesn't change. So here, if we have cracked the first column to the second one, we obtained the Matrix a B minus a C 101 who has the same determinant as the original Matrix. Andi. If we subtract again the first column to the third column, then we obtained the Matrix A B minus a C minus a 100 on DFO. Um, here. If we use miners ankle factories along the third or last wrote, we can see that this determinant is the same as a determinant of B minus A and C minus a, um, the matrix. Who that has those columns on. We knew from before that this determinant is twice the area of the triangle A region are determined by a V NC. So from here we can conclude that, uh, if we multiply this determinant off the matrix whose columns are the on the unions forms of avian see, we multiply this determinant, but I won't have what we obtain. ISS the area off the triangle, a region determined by a V insee

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