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Chapter 8: Q17E (page 437)
In Exercises 17–20, prove the given statement about subsets \(A\) and \(B\) of \({R^n}\) . A proof for an exercise may use results of earlier exercises.
17. If \(A \subset B\) and \(B\) is convex, then \({\rm{conv}}\,A \subset B\).
It is shown that \({\rm{conv}}\,A \subset B\).
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Let\(T\)be a tetrahedron in “standard” position, with three edges along the three positive coordinate axes in\({\mathbb{R}^3}\), and suppose the vertices are\(a{{\bf{e}}_1}\),\(b{{\bf{e}}_2}\),\(c{{\bf{e}}_{\bf{3}}}\), and 0, where\(\left[ {\begin{array}{*{20}{c}}{{{\bf{e}}_1}}&{{{\bf{e}}_2}}&{{{\bf{e}}_3}}\end{array}} \right] = {I_3}\). Find formulas for the barycentric coordinates of an arbitrary point\({\bf{p}}\)in\({\mathbb{R}^3}\).
In Exercises 1-4, write y as an affine combination of the other point listed, if possible.
\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{2}}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{4}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{aligned}} \right)\)
Let \({\bf{x}}\left( t \right)\) be a cubic Bézier curve determined by points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\), \({{\bf{p}}_2}\), and \({{\bf{p}}_3}\).
a. Compute the tangent vector \({\bf{x}}'\left( t \right)\). Determine how \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have \({\bf{x}}'\left( 1 \right) = 0\)?
b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_1}\) as the origin of the coordinate system.]
Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{0}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).
Question: 23. Let \({{\bf{v}}_1} = \left( \begin{array}{l}1\\1\end{array} \right)\), \({{\bf{v}}_2} = \left( \begin{array}{l}3\\0\end{array} \right)\), \({{\bf{v}}_3} = \left( \begin{array}{l}5\\3\end{array} \right)\) and \({\bf{p}} = \left( \begin{array}{l}4\\1\end{array} \right)\). Find a hyperplane \(f:d\) (in this case, a line) that strictly separates \({\bf{p}}\) from \({\rm{conv}}\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\).
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