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Chapter 8: Q8.1-17E (page 437)

Question: 17. Choose a set \(S\) of three points such that aff \(S\) is the plane in \({\mathbb{R}^3}\) whose equation is \({x_3} = 5\). Justify your work.

Short Answer

Expert verified

The set is \(S = \left\{ {\left( \begin{array}{l}0\\0\\5\end{array} \right),\left( \begin{array}{l}1\\0\\5\end{array} \right),\left( \begin{array}{l}1\\1\\5\end{array} \right)} \right\}\).

Step by step solution

01

Describe the given statement

The set of three vectors that lie along the plane \({x_3} = 5\) must have 5 as their third entry and cannot be collinear.

The set of vectors that is not collinear cannot have a line as their affine hull.

02

 Draw a conclusion

One of the possible sets of three vectors discussed above is \(S = \left\{ {\left( \begin{array}{l}0\\0\\5\end{array} \right),\left( \begin{array}{l}1\\0\\5\end{array} \right),\left( \begin{array}{l}1\\1\\5\end{array} \right)} \right\}\).

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Most popular questions from this chapter

Question: Let \({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{ - {\bf{1}}}\\{\bf{3}}\end{array}} \right)\), \({{\bf{n}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{4}}\\{\bf{2}}\end{array}} \right)\), and \({{\bf{n}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\\{\bf{1}}\\{\bf{5}}\end{array}} \right)\), let \({H_{\bf{1}}}\) be the hyperplane in \({\mathbb{R}^{\bf{4}}}\) through \({{\bf{p}}_{\bf{1}}}\) with normal \({{\bf{n}}_{\bf{1}}}\), and let \({H_{\bf{2}}}\) be the hyperplane through \({{\bf{p}}_{\bf{2}}}\) with normal \({{\bf{n}}_{\bf{2}}}\). Give an explicit description of \({H_{\bf{1}}} \cap {H_{\bf{2}}}\). (Hint: Find a point p in \({H_{\bf{1}}} \cap {H_{\bf{2}}}\) and two linearly independent vectors \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) that span a subspace parallel to the 2-dimensional flat \({H_{\bf{1}}} \cap {H_{\bf{2}}}\).)

Let \(W = \left\{ {{{\bf{v}}_1},......,{{\bf{v}}_p}} \right\}\). Show that if \({\bf{x}}\) is orthogonal to each \({{\bf{v}}_j}\), for \(1 \le j \le p\), then \({\bf{x}}\) is orthogonal to every vector in \(W\).

Question: 18. Choose a set \(S\) of four points in \({\mathbb{R}^3}\) such that aff \(S\) is the plane \(2{x_1} + {x_2} - 3{x_3} = 12\). Justify your work.

In Exercises 21鈥�24, a, b, and c are noncollinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \)and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

24. Take q on the line segment from b to c and consider the line through q and a, which may be written as\(p = \left( {1 - x} \right)q + xa\)for all real x. Show that, for each x,\(det\left[ {\begin{array}{*{20}{c}}{\widetilde p}&{\widetilde b}&{\widetilde c}\end{array}} \right] = x \cdot det\left[ {\begin{array}{*{20}{c}}{\widetilde a}&{\widetilde b}&{\widetilde c}\end{array}} \right]\). From this and earlier work, conclude that the parameter x is the first barycentric coordinate of p. However, by construction, the parameter x also determines the relative distance between p and q along the segment from q to a. (When x = 1, p = a.) When this fact is applied to Example 5, it shows that the colors at vertex a and the point q are smoothly interpolated as p moves along the line between a and q.

Question: Repeat Exercise 7 when

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{3}}\\{ - {\bf{2}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{6}}\\{ - {\bf{5}}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{0}}\\{{\bf{12}}}\\{ - {\bf{6}}}\end{array}} \right)\)

\({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{1}}}\\{{\bf{15}}}\\{ - {\bf{7}}}\end{array}} \right)\), \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{5}}}\\{\bf{3}}\\{ - {\bf{8}}}\\{\bf{6}}\end{array}} \right)\), and \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{6}}\\{ - {\bf{6}}}\\{ - {\bf{8}}}\end{array}} \right)\)
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