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Chapter 3: Q42Q (page 165)

Let \(u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\), and \(v = \left[ {\begin{aligned}{*{20}{c}}c\\{\bf{0}}\end{aligned}} \right]\), where a, b, and c are positive integers (for simplicity). Compute the area of the parallelogram determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right]\), and \[\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\]. Draw a picture and explain what you find.

Short Answer

Expert verified

The area of the parallelogram is \(cb\) square units.

\({\rm{det}}\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right] = - cb\), and \({\rm{det}}\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right] = cb\).

Step by step solution

01

Write the vector

Obtain the vector \(u + v\)using the vectors \[u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\]and \(v = \left[ {\begin{aligned}{*{20}{c}}c\\0\end{aligned}} \right]\).

\(\begin{aligned}{c}u + v = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right] + \left[ {\begin{aligned}{*{20}{c}}c\\0\end{aligned}} \right]\\ = \left[ {\begin{aligned}{*{20}{c}}{a + c}\\{b + 0}\end{aligned}} \right]\\ = \left[ {\begin{aligned}{*{20}{c}}{a + c}\\b\end{aligned}} \right]\end{aligned}\)

Thus, the vector is \(u + v = \left[ {\begin{aligned}{*{20}{c}}{a + c}\\b\end{aligned}} \right]\).

02

Display the vectors on the graph

The graph of vectors \[u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\], \(v = \left[ {\begin{aligned}{*{20}{c}}c\\0\end{aligned}} \right]\), \(u + v = \left[ {\begin{aligned}{*{20}{c}}{a + c}\\b\end{aligned}} \right]\), and 0 using the arrows is shown below:

Here, the length of the base of the parallelogram is c units, and the height is b units.

The area of the parallelogram is calculated below:

\(\begin{aligned}{c}{\rm{Area}} = c \times b\\ = cb\end{aligned}\)

Thus, the area is \(cb\) square units.

03

Compute the determinant

Obtain the determinant of vectors\(\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right]\)and\(\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\).

\(\begin{aligned}{c}{\rm{det}}\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right] = \left| {\begin{aligned}{*{20}{c}}a&c\\b&0\end{aligned}} \right|\\ = a\left( 0 \right) - c\left( b \right)\\ = - cb\end{aligned}\)

Thus,\({\rm{det}}\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right] = - cb\).

And

\(\begin{aligned}{c}{\rm{det}}\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right] = \left| {\begin{aligned}{*{20}{c}}c&a\\0&b\end{aligned}} \right|\\ = c\left( b \right) - a\left( 0 \right)\\ = cb\end{aligned}\)

Thus,\({\rm{det}}\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\).

The area computed by the graph and by the determinant of vectors is the same.

It means the sides of the parallelogram adjacent to 0 defines the side of the parallelogram, and it is equal to the area of the parallelogram.

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Most popular questions from this chapter

Question:In Exercises 31鈥�36, mention an appropriate theorem in your explanation.

36. Let U be a square matrix such that \({U^T}U = I\). Show that\(det{\rm{ }}U = \pm 1\).

In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right]\).

33. \(\left[ {\begin{aligned}{*{20}{c}}1&k\\0&1\end{aligned}} \right]\)

Compute the determinants of the elementary matrices given in Exercises 25-30. (See Section 2.2)

\[\left[ {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}&{\bf{0}}\\{\bf{0}}&k&{\bf{1}}\end{aligned}} \right]\]

Find the determinant in Exercise 16, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].

16. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{{\bf{5d}}}&{{\bf{5e}}}&{{\bf{5f}}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right|\]

Let \(u = \left[ {\begin{array}{*{20}{c}}3\\0\end{array}} \right]\), and \(v = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Compute the area of the parallelogram

determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{array}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{array}} \right]\). How do they compare? Replace the first entry of v by an arbitrary number x, and repeat the problem. Draw a picture and explain what you find.
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