Linear Algebra: A Modern Introduction

Published by Cengage Learning
ISBN 10: 1285463242
ISBN 13: 978-1-28546-324-7

Chapter 1 - Vectors - 1.3 Lines and Planes - Exercises 1.3 - Page 44: 9


Vector form:$\begin{bmatrix}{x \\y \\z}\end{bmatrix} = s\begin{bmatrix}{2 \\1\\2}\end{bmatrix}+t\begin{bmatrix}{-3 \\2\\1}\end{bmatrix}$ Parametric form: $x=2s-3t, y=s+2t,z=2s+t$

Work Step by Step

The vector form of a line is: $x=p+su+tv$ This implies, $\begin{bmatrix}{x \\y \\z}\end{bmatrix} = \begin{bmatrix}{0 \\0 \\0}\end{bmatrix}+s\begin{bmatrix}{2 \\1\\2}\end{bmatrix}+t\begin{bmatrix}{-3 \\2\\1}\end{bmatrix}$ or, $\begin{bmatrix}{x \\y \\z}\end{bmatrix} = s\begin{bmatrix}{2 \\1\\2}\end{bmatrix}+t\begin{bmatrix}{-3 \\2\\1}\end{bmatrix}$ Parametric equations of a line are defined as such equations which correspond to the components of the vector. Thus, the parametric form of the equation of a line is: $x=2s-3t, y=s+2t,z=2s+t$ Hence, the vector form is:$\begin{bmatrix}{x \\y \\z}\end{bmatrix} = s\begin{bmatrix}{2 \\1\\2}\end{bmatrix}+t\begin{bmatrix}{-3 \\2\\1}\end{bmatrix}$ The parametric form is: $x=2s-3t, y=s+2t,z=2s+t$
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