Chapter 8.8, Problem 5CFU

(a)

To determine

To Find: The matrix of the given figure.

Answer to Problem 5CFU

The required matrix is $M=\left[\begin{array}{cccccccc}5& 5& 0& 0& 0& 5& 5& 0\\ 2& 5& 5& 2& 2& 2& 5& 5\\ 0& 0& 0& 0& 4& 4& 4& 4\end{array}\right]$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

Figure 1

Calculation:

Consider the vertices of the graph is,

  $\begin{array}{l}A\left(5,5-3,0\right)=A\left(5,2,0\right)\\ C\left(5-5,5,0\right)=C\left(0,5,0\right)\\ D\left(0,2,4-4\right)=D\left(0,2,0\right)\\ F\left(0+5,2,4\right)=F\left(5,2,4\right)\end{array}$

Also,

  $\begin{array}{l}G\left(0,+5,2+3,\right)=G\left(5,5,4\right)\\ H\left(0,2+3,4\right)=H\left(0,5,4\right)\end{array}$

Then, the matrix of the figure is shown below,

  $M=\left[\begin{array}{cccccccc}5& 5& 0& 0& 0& 5& 5& 0\\ 2& 5& 5& 2& 2& 2& 5& 5\\ 0& 0& 0& 0& 4& 4& 4& 4\end{array}\right]$

(b)

To determine

To Find: Theresulting matrix when the figure is translate the figure by the vector $\left(4,1,2\right)$ .

Answer to Problem 5CFU

The graph of the function is shown in Figure 1

Explanation of Solution

Consider the vector is $\left(4,1,2\right)$ then the matrix is,

  $T\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$

Consider the diagram shown in Figure 2

Figure 2

The transformation of the matrix along the vector $\left(4,1,2\right)$ is obtained as,

  $\begin{array}{c}\left[\begin{array}{cccccccc}4& 4& 4& 4& 4& 4& 4& 4\\ -1& -1& -1& -1& -1& -1& -1& -1\\ 2& 2& 2& 2& 2& 2& 2& 2\end{array}\right]+\left[\begin{array}{cccccccc}5& 5& 0& 0& 0& 5& 5& 0\\ 2& 5& 5& 2& 2& 2& 5& 5\\ 0& 0& 0& 0& 4& 4& 4& 4\end{array}\right]=\\ \left[\begin{array}{cccccccc}9& 9& 4& 4& 4& 9& 9& 4\\ 1& 4& 4& 1& 1& 1& 4& 4\\ 2& 2& 2& 2& 6& 6& 6& 6\end{array}\right]\end{array}$

(c)

To determine

To Find: The transformation of the given figure by the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$ and then graph the image and then describe the result.

Explanation of Solution

Consider the matrix is $\left[\begin{array}{cccccccc}9& 9& 4& 4& 4& 9& 9& 4\\ 1& 4& 4& 1& 1& 1& 4& 4\\ 2& 2& 2& 2& 6& 6& 6& 6\end{array}\right]$ .

The transformation of the given figure by the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$ is shown in Figure 3

Figure 3

(d)

To determine

To Find: The transformation of the Figure that result from the product with the matrix $\left[\begin{array}{ccc}0.5& 0& 0\\ 0& 0.5& 0\\ 0& 0& 0.5\end{array}\right]$ .

Answer to Problem 5CFU

The dimension of the prism is increased by the factor of 0.5.

Explanation of Solution

From Figure 3 transformation of the Figure that result from the product with the matrix $\left[\begin{array}{ccc}0.5& 0& 0\\ 0& 0.5& 0\\ 0& 0& 0.5\end{array}\right]$ result in the increase in dimensions of the prism that is increased by the factor of 0.5.