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Chapter 1: Q20E (page 1)

In Exercises 19 and 20, find the parametric equation of the line

through a parallel to b.

20. \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}3\\{ - 4}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{ - 7}\\8\end{array}} \right]\)

Short Answer

Expert verified

The parametric equations are \({x_1} = 3 - 7t\) and \({x_2} = - 4 + 8t\).

Step by step solution

01

Write the general parametric equation of the line

If a line passes through vector\({\bf{a}}\)and is parallel to vector b,then the parametric equation of the line is represented as\({\bf{x}} = {\bf{a}} + t{\bf{b}}\), where\(t\)is a parameter.

Here, \({\bf{x}}\) is represented as \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).

02

Substitute the vectors in the parametric equations

Consider the parametric equation \({\bf{x}} = {\bf{a}} + t{\bf{b}}\).

Substitute the vectors\({\bf{a}} = \left[ {\begin{array}{*{20}{c}}3\\{ - 4}\end{array}} \right]\)and\({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{ - 7}\\8\end{array}} \right]\) in the equation \({\bf{x}} = {\bf{a}} + t{\bf{b}}\)as shown below:

\({\bf{x}} = \left[ {\begin{array}{*{20}{c}}3\\{ - 4}\end{array}} \right] + t\left[ {\begin{array}{*{20}{c}}{ - 7}\\8\end{array}} \right]\)

Thus, \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}3\\{ - 4}\end{array}} \right] + t\left[ {\begin{array}{*{20}{c}}{ - 7}\\8\end{array}} \right]\).

03

Equate the vectors

Substitute \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\) in the above equation.

\(\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3\\{ - 4}\end{array}} \right] + t\left[ {\begin{array}{*{20}{c}}{ - 7}\\8\end{array}} \right]\)

Simplify further.

\(\begin{array}{l}\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3\\{ - 4}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ - 7t}\\{8t}\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3 - 7t}\\{ - 4 + 8t}\end{array}} \right]\end{array}\)

04

Obtain the parametric equations of the line

Equate the vectors \(\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3 - 7t}\\{ - 4 + 8t}\end{array}} \right]\) to obtain the parametric equations.

Thus, the parametric equations are \({x_1} = 3 - 7t\) and \({x_2} = - 4 + 8t\).

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Most popular questions from this chapter

Suppose Ais an \(n \times n\) matrix with the property that the equation \(A{\mathop{\rm x}\nolimits} = 0\) has at least one solution for each b in \({\mathbb{R}^n}\). Without using Theorem 5 or 8, explain why each equation Ax = b has in fact exactly one solution.

Find the general solutions of the systems whose augmented matrices are given

11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).

In Exercise 23 and 24, make each statement True or False. Justify each answer.

24.

a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).

b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).

c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.

d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.

e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).

Let \(u = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\) and \(v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\). Show that \(\left[ {\begin{array}{*{20}{c}}h\\k\end{array}} \right]\) is in Span \(\left\{ {u,v} \right\}\) for all \(h\) and\(k\).

Solve the linear system of equations. You may use technology.

|3x+5y+3z=257X+9y+19z=654X+5y+11z=5|
See all solutions

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