Question

In Exercises 21 and 22, find a parametric equation of the line \(M\) through \({\mathop{\rm p}\nolimits} \) and \({\mathop{\rm q}\nolimits} \). [Hint: \(M\) is parallel to the vector \({\mathop{\rm q}\nolimits} - p\). See the figure below.]

22. \({\mathop{\rm p}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right]\), \(q = \left[ {\begin{array}{*{20}{c}}0\\{ - 4}\end{array}} \right]\)

Short answer

The parametric equation of line \(M\) through \(p\) and \(q\) is \(x = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right] + t\left[ {\begin{array}{*{20}{c}}6\\{ - 7}\end{array}} \right].\)

Step-by-step solution

Observation from the given figure

Line \(M\) passing through \(p\) and \(q\) is parallel to the vector \({\mathop{\rm q}\nolimits} - {\mathop{\rm p}\nolimits} \).

Use the given part to solve the vector \(q - p\) 

It is given that\({\mathop{\rm p}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right]\)and\({\mathop{\rm q}\nolimits} = \left[ {\begin{array}{*{20}{c}}0\\{ - 4}\end{array}} \right]\).

\(\begin{array}{c}q - p = \left[ {\begin{array}{*{20}{c}}0\\{ - 4}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0 + 6}\\{ - 4 - 3}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}6\\{ - 7}\end{array}} \right]\end{array}\)

Determine the parametric equation of line \(M\)

It is known that the equation of the linethrough \(p\) parallel to \({\mathop{\rm v}\nolimits} \) is represented by

\(x = {\mathop{\rm p}\nolimits} + t{\mathop{\rm v}\nolimits} \left( {t\,{\mathop{\rm in}\nolimits} \,\mathbb{R}} \right)\). The solution set of \(Ax = {\mathop{\rm b}\nolimits} \) is a line through \({\mathop{\rm p}\nolimits} \) parallel to the solution set \(Ax = 0\).

Write the parametric equation of line \(M\) parallel to vector \({\mathop{\rm q}\nolimits} - p\) as:

\(\begin{array}{c}x = p + t\left( {q - p} \right)\\ = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right] + t\left[ {\begin{array}{*{20}{c}}6\\{ - 7}\end{array}} \right]\end{array}\)

Thus, the parametric equation of line \(M\) through \(p\) and \(q\) is \(x = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right] + t\left[ {\begin{array}{*{20}{c}}6\\{ - 7}\end{array}} \right]\).