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Chapter 8: Q10E (page 437)

Repeat Exercise 9 for the points \({{\bf{q}}_{\bf{1}}}\),….\({{\bf{q}}_{\bf{5}}}\) whose barycentric coordinates with respect to S are given by \(\left( {\frac{{\bf{1}}}{{\bf{8}}},\frac{{\bf{1}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{8}}},\frac{{\bf{1}}}{{\bf{2}}}} \right)\), \(\left( {\frac{{\bf{3}}}{{\bf{4}}}, - \frac{{\bf{1}}}{{\bf{4}}},{\bf{0}},\frac{{\bf{1}}}{{\bf{2}}}} \right)\),\(\left( {{\bf{0}},\frac{{\bf{3}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{4}}},{\bf{0}}} \right)\),\(\left( {{\bf{0}}, - {\bf{2}},{\bf{0}},{\bf{3}}} \right)\), and \(\left( {\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{3}}},{\bf{0}}} \right)\), respectively.

Short Answer

Expert verified

\({{\bf{q}}_1}\) is inside the tetrahedron \({\rm{conv}}\,S\), \({{\bf{q}}_2}\) is outside \({\rm{conv}}\,S\), \({{\bf{q}}_3}\) is the edge between \({{\bf{v}}_2}\) and \({{\bf{v}}_3}\). \({{\bf{q}}_4}\) is outside the tetrahedron and \({{\bf{q}}_5}\) is on the face with vertices \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_3}\).

Step by step solution

01

Check for the first coordinate

The barycentric coordinateof the point \({{\bf{q}}_1}\) are all positive, so the point \({{\bf{q}}_1}\) is inside the tetrahedron convex S.

02

Check for the second coordinate

The barycentric coordinate of the point \({{\bf{q}}_2}\)is not positive, so the point \({{\bf{q}}_2}\) is outside the tetrahedron convexS.

03

Check for the third coordinate

The first and fourth barycentric coordinate of the point \({{\bf{q}}_3}\)is zero, representing the edge between \({{\bf{v}}_2}\) and \({{\bf{v}}_3}\).

04

Check for the fourth coordinate

The barycentric coordinate of the point \({{\bf{q}}_4}\)is not positive, so the point \({{\bf{q}}_4}\) is outside the tetrahedron convex S.

05

Check for the fifth coordinate

The point \({{\bf{q}}_5}\) is a convex combination of vectors \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_3}\), hence \({{\bf{q}}_5}\) lies on face containing the vertices \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_3}\).

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Most popular questions from this chapter

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{ - {\bf{6}}}\\{\bf{7}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}\\{\bf{3}}\\{\bf{1}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{3}}}\\{\bf{4}}\\{ - {\bf{4}}}\end{aligned}} \right)\)

Suppose that\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)is an affinely independent set in\({\mathbb{R}^{\bf{n}}}\)and q is an arbitrary point in\({\mathbb{R}^{\bf{n}}}\). Show that the translated set\(\left\{ {{p_1} + q,{p_2} + q,{p_3} + {\bf{q}}} \right\}\)is also affinely independent.

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\3\\{ - 6}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{7}}\\3\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{9}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{9}}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{1.4}\\{{\bf{1}}.{\bf{5}}}\\{ - {\bf{3}}.{\bf{1}}}\end{array}} \right]\), and \({\bf{x}}\left( t \right) = {\bf{a}} + t{\bf{b}}\)for \(t \ge {\bf{0}}\).Find the point where the ray\({\bf{x}}\left( t \right)\)intersects the plane that contains the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). Is this point inside the triangle?

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

3.\(\left( {\begin{aligned}{{}}1\\2\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 2}\\{ - 4}\\8\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\{ - 1}\\{11}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\{15}\\{ - 9}\end{aligned}} \right)\)

Repeat Exercise 25 with\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\{\bf{2}}\\{ - {\bf{4}}}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{8}}\\{\bf{2}}\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{{\bf{10}}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{8}}\end{array}} \right]\), and \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{.{\bf{9}}}\\{{\bf{2}}.{\bf{0}}}\\{ - {\bf{3}}.{\bf{7}}}\end{array}} \right]\).
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