Chapter EP, Problem 4.4.2EP

(a)

To determine

To Find: The coordinate matrix for $\Delta MQN$ is the vertex matrix.

Answer to Problem 4.4.2EP

The required matrix is $\left[\begin{array}{ccc}2& 3& 1\\ 4& -5& -1\end{array}\right]$ .

Explanation of Solution

Given:

The vertices of the triangle MQN are $M\left(2,4\right)Q\left(3,-5\right)N\left(1,-1\right)$ .

Calculation:

Consider the given vertices is $M\left(2,4\right)Q\left(3,-5\right)N\left(1,-1\right)$ .

The required matrix for the coordinates is,

  $MQN=\left[\begin{array}{ccc}2& 3& 1\\ 4& -5& -1\end{array}\right]$

(b)

To determine

To Find: The reflection matrix for reflecting over the line $y=x$ .

Answer to Problem 4.4.2EP

The value of the reflection matrix is $\left[\begin{array}{ccc}4& -5& -1\\ 2& 3& 1\end{array}\right]$ .

Explanation of Solution

Consider the vertex matrix is,

  $MQN=\left[\begin{array}{ccc}2& 3& 1\\ 4& -5& -1\end{array}\right]$

The matrix after reflecting over $y=x$ is,

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}2& 3& 1\\ 4& -5& -1\end{array}\right]=\left[\begin{array}{ccc}4& -5& -1\\ 2& 3& 1\end{array}\right]$

(c)

To determine

To Find: The coordinates of the reflected matrix.

Answer to Problem 4.4.2EP

The coordinates of the reflected matrix is $M^{\prime }\left(4,2\right)Q^{\prime }\left(-5,3\right)N^{\prime }\left(-1,1\right)$ .

Explanation of Solution

Consider the reflected matrix is,

  $\left[\begin{array}{ccc}4& -5& -1\\ 2& 3& 1\end{array}\right]$

Then, the coordinates of the reflected matrix is $M^{\prime }\left(4,2\right)Q^{\prime }\left(-5,3\right)N^{\prime }\left(-1,1\right)$ .

(d)

To determine

To Find: The coordinates of the vertical matrix.

Answer to Problem 4.4.2EP

The graph for the two figures is $MQN$ and $M^{\prime }{Q}^{\prime }{N}^{\prime }$ .

Explanation of Solution

The required graph for the $MQN$ and $M^{\prime }{Q}^{\prime }{N}^{\prime }$ is shown in Figure 1

  

Figure 1

(e)

To determine

To Find: The rotation matrix for rotating $\Delta MQN$ 90 degree counter clockwise about the origin.

Answer to Problem 4.4.2EP

The required matrix is $\left[\begin{array}{ccc}-4& 5& 1\\ 2& 3& 1\end{array}\right]$ .

Explanation of Solution

The required matrix for the coordinates is,

  $MQN=\left[\begin{array}{ccc}2& 3& 1\\ 4& -5& -1\end{array}\right]$

The rotation matrix for rotating $\Delta MQN$ 90 degree counter clockwise is obtained as,

  $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}2& 3& 1\\ 4& -5& -1\end{array}\right]=\left[\begin{array}{ccc}-4& 5& 1\\ 2& 3& 1\end{array}\right]$

(f)

To determine

To Find: The coordinates of the rotated matrix.

Answer to Problem 4.4.2EP

The coordinates of the rotated matrix is $M^{\prime }\left(-4,2\right)Q^{\prime }\left(5,3\right)N^{\prime }\left(1,1\right)$ .

Explanation of Solution

Consider the rotated matrix is,

  $\left[\begin{array}{ccc}-4& 5& 1\\ 2& 3& 1\end{array}\right]$

Then, the coordinates of the rotated matrix is $M^{\prime }\left(-4,2\right)Q^{\prime }\left(5,3\right)N^{\prime }\left(1,1\right)$ .

(g)

To determine

To Find: The graph of the original and the rotated matrix.

Answer to Problem 4.4.2EP

The coordinates of the rotated matrix is $M^{\prime }\left(-4,2\right)Q^{\prime }\left(5,3\right)N^{\prime }\left(-1,1\right)$ .

Explanation of Solution

The coordinates of the rotated figure is $M^{\prime }\left(-4,2\right)Q^{\prime }\left(5,3\right)N^{\prime }\left(-1,1\right)$ and the original figure is $M\left(2,4\right)Q\left(3,-5\right)N\left(1,-1\right)$

The graph for the figure is shown in Figure 2

  

Figure 2