Q21E Let ${{\bf{p}}_o}$ , \({{\bf{p... [FREE SOLUTION]

91影视

Linear Algebra and its Applications

David C. Lay, Steven R. Lay and Judi J. McDonald

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 483 pages

ISBN: 978-03219822384

Chapter 8: Q21E (page 437)

Let ${{\bf{p}}_o}$ , ${{\bf{p}}_1}$ and ${{\bf{p}}_2}$ be points in ${\mathbb{R}^n}$ , and define ${{\bf{f}}_{\bf{0}}}\left( t \right) = \left( {1 - t} \right){{\bf{p}}_{\bf{0}}} + t{{\bf{p}}_{\bf{1}}}$, ${{\bf{f}}_1}\left( t \right) = \left( {1 - t} \right){{\bf{p}}_1} + t{{\bf{p}}_2}$ and ${\bf{g}}\left( t \right) = \left( {1 - t} \right){{\bf{f}}_0}\left( t \right) + t{{\bf{f}}_1}\left( t \right)$for $0 \le t \le 1$. For the points as shown below, draw a picture that shows ${{\bf{f}}_0}\left( {\frac{1}{2}} \right)$, ${{\bf{f}}_1}\left( {\frac{1}{2}} \right)$, and ${\bf{g}}\left( {\frac{1}{2}} \right)$.

Short Answer

Expert verified

The diagram is shown below:

Step by step solution

Describe the given information

It is given that ${{\rm{p}}_0},{{\rm{p}}_1},{{\rm{p}}_2} \in {\mathbb{R}^n}$. Also,${{\rm{f}}_0}\left( t \right) = \left( {1 - t} \right){{\rm{p}}_0} + t{{\rm{p}}_1}$, ${{\rm{f}}_1}\left( t \right) = \left( {1 - t} \right){{\rm{p}}_1} + t{{\rm{p}}_2}$and ${\rm{g}}\left( t \right) = \left( {1 - t} \right){{\rm{f}}_0}\left( t \right) + t{{\rm{f}}_1}\left( t \right)$.The graph of${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$,${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$ and${\rm{g}}\left( {\frac{1}{2}} \right)$ is to be drawn.

Step 2:Find the values of${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$,${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$ and${\rm{g}}\left( {\frac{1}{2}} \right)$

The values of ${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$,${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$ and${\rm{g}}\left( {\frac{1}{2}} \right)$are calculated as shown below:

$\begin{aligned}{}{{\rm{f}}_0}\left( {\frac{1}{2}} \right) &= \left( {1 - \frac{1}{2}} \right){{\rm{p}}_0} + \frac{1}{2}{{\rm{p}}_1}\\ &= \frac{1}{2}{{\rm{p}}_0} + \frac{1}{2}{{\rm{p}}_1}\\ &= \frac{1}{2}\left( {{{\rm{p}}_0} + {{\rm{p}}_1}} \right)\end{aligned}$

It shows that${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$is at the midline formed by the points${{\rm{p}}_0}{\rm{, }}{{\rm{p}}_1}$.

$\begin{aligned}{}{{\rm{f}}_1}\left( {\frac{1}{2}} \right) &= \left( {1 - \frac{1}{2}} \right){{\rm{p}}_1} + \frac{1}{2}{{\rm{p}}_2}\\ &= \frac{1}{2}{{\rm{p}}_1} + \frac{1}{2}{{\rm{p}}_2}\\ &= \frac{1}{2}\left( {{{\rm{p}}_1} + {{\rm{p}}_2}} \right)\end{aligned}$

It shows that ${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$is at the midline formed by the points${{\rm{p}}_1}{\rm{, }}{{\rm{p}}_2}$.

$\begin{aligned}{}{\rm{g}}\left( {\frac{1}{2}} \right) &= \left( {1 - \frac{1}{2}} \right){{\rm{f}}_0}\left( {\frac{1}{2}} \right) + \frac{1}{2}{{\rm{f}}_1}\left( {\frac{1}{2}} \right)\\ &= \frac{1}{2}{{\rm{f}}_0}\left( {\frac{1}{2}} \right) + \frac{1}{2}{{\rm{f}}_1}\left( {\frac{1}{2}} \right)\\\frac{1}{2}\left( {{{\rm{f}}_0}\left( {\frac{1}{2}} \right) + {{\rm{f}}_1}\left( {\frac{1}{2}} \right)} \right)\end{aligned}$

It shows that ${\rm{g}}\left( {\frac{1}{2}} \right)$ is at the midline formed by the points of ${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$,${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$.

Step 3:Draw the diagram

Draw lines joining${{\rm{p}}_0}{\rm{, }}{{\rm{p}}_1}$, then${{\rm{p}}_1}{\rm{, }}{{\rm{p}}_2}$,such that${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$,${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$bisects them respectively. Afterward, the value of${\rm{g}}\left( {\frac{1}{2}} \right)$is at the mid of the line joining${{\rm{f}}_0}\left( {\frac{1}{2}} \right)$,${{\rm{f}}_1}\left( {\frac{1}{2}} \right)$. Thus, the diagram is shown below: