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Chapter 8: Q25E (page 437)

Question: 25. Let \(p = \left( \begin{array}{l}1\\1\end{array} \right)\). Find a hyperplane \(\left( {f:d} \right)\) that strictly separates \(B\left( {0,3} \right)\) and \(B\left( {p,1} \right)\). (Hint: After finding \(f\), show that the point \(v = \left( {1 - .75} \right)0 + .75p\) is neither in \(B\left( {0,3} \right)\) nor in \(B\left( {p,1} \right)\)).

Short Answer

Expert verified

The hyperplane is \(f\left( {x:d} \right) = f\left( {x:n \cdot x = 12.75} \right)\).

Step by step solution

01

Find the hyper plane with normal vector \(p\)

Consider the length of the line \(L\) joining thecentercentreof \(B\left( {0,3} \right)\) and \(B\left( {p,1} \right)\) is \(\sqrt {{4^2} + {1^2}} = 4.12\).

This distance is greater than the overall radii of the balls. So, thedisk areisuntouched.

This does not require the hyperplane to be normaltoto \(L\). So, a hyperplane whose normal vector is p would work. The hyperplane \(H\) with normal vector \(p\) is \(f\left( x \right) = p \cdot x\).

02

Find a point \(q\) on the line \(L\)

As the radii of the balls are 3 and 1, the point \(d\) should be \(\frac{3}{4}\) of the distance from the centers of the ball. So, obtain a point \(q\) at the distance ratio of \(\frac{3}{4}\) on line \(L\) as shown below:

\(\begin{array}{c}q = \left( {.25} \right)0 + \left( {0.75} \right)p\\ = \left( {3,\,\,.75} \right)\end{array}\)

03

Evaluate \(d\)as \(d = f\left( q \right)\)

The real number d is shown below:

\(\begin{array}{c}d = f\left( q \right)\\ = p \cdot q\\ = 4\left( 3 \right) + 1\left( {.75} \right)\\ = 12.75\end{array}\)

04

Find the hyperplane \(f\left( {x:d} \right)\)

The required hyper plane is \(f\left( {x:n \cdot x = 12.75} \right)\).

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Most popular questions from this chapter

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.

2.\(\left( {\begin{aligned}{{}}2\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}5\\4\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 3}\\{ - 2}\end{aligned}} \right)\)

Questions: Let \({F_{\bf{1}}}\) and \({F_{\bf{2}}}\) be 4-dimensional flats in \({\mathbb{R}^{\bf{6}}}\), and suppose that \({F_{\bf{1}}} \cap {F_{\bf{2}}} \ne \phi \). What are the possible dimension of \({F_{\bf{1}}} \cap {F_{\bf{2}}}\)?

In Exercises 21–24, a, b, and c are non-collinear 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.

22. Let p be a point on the line through a and b. Show that\(det\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{p}} }\end{array}} \right] = 0\).

In Exercises 7 and 8, find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.

8. \(\left( {\begin{array}{{}}0\\1\\{ - 2}\\1\end{array}} \right),\left( {\begin{array}{{}}1\\1\\0\\2\end{array}} \right),\left( {\begin{array}{{}}1\\4\\{ - 6}\\5\end{array}} \right)\), \({\mathop{\rm p}\nolimits} = \left( {\begin{array}{{}}{ - 1}\\1\\{ - 4}\\0\end{array}} \right)\)

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\({\mathbb{R}^3}\).
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