Chapter 8.8, Problem 20E

a.

To determine

To find: the translation matrix used for transformation.

Answer to Problem 20E

  $\left[\begin{array}{rrrrr}\hfill 23.6& \hfill 23.6& \hfill 23.6& \hfill 23.6& \hfill 23.6\\ \hfill 72& \hfill 72& \hfill 72& \hfill 72& \hfill 72\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$

Explanation of Solution

Given information:

The matrix representing the group of dolphins with the ship at the origin is shown below.

  $\left[\begin{array}{rrrrr}\hfill 20& \hfill 136& \hfill 247& \hfill 302& \hfill 351\\ \hfill -58& \hfill -71& \hfill -74& \hfill -83& \hfill -62\\ \hfill 27& \hfill 53& \hfill 59& \hfill 37& \hfill 52\end{array}\right]$

Later the matrix is translated into a fixed reference point using the vector $\left(23.6,72,0\right)$ .

Calculation:

The translation matrix will add 23.6 to each $x$ -coordinate and 72 to each $y$ -coordinate. The $z$ -coordinate of the transformation matrix will remain same, so the translation matrix will be as follows:

  $\left[\begin{array}{rrrrr}\hfill 23.6& \hfill 23.6& \hfill 23.6& \hfill 23.6& \hfill 23.6\\ \hfill 72& \hfill 72& \hfill 72& \hfill 72& \hfill 72\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$

b.

To determine

To find: the resulting matrix.

Answer to Problem 20E

  $\left[\begin{array}{rrrrr}\hfill 43.6& \hfill 159.6& \hfill 270.6& \hfill 325.6& \hfill 374.6\\ \hfill 14& \hfill 1& \hfill -2& \hfill -11& \hfill 10\\ \hfill 27& \hfill 53& \hfill 59& \hfill 37& \hfill 52\end{array}\right]$

Explanation of Solution

Given information:

The matrix representing the group of dolphins with the ship at the origin is shown below.

  $\left[\begin{array}{rrrrr}\hfill 20& \hfill 136& \hfill 247& \hfill 302& \hfill 351\\ \hfill -58& \hfill -71& \hfill -74& \hfill -83& \hfill -62\\ \hfill 27& \hfill 53& \hfill 59& \hfill 37& \hfill 52\end{array}\right]$

Later the matrix is translated into a fixed reference point using the vector $\left(23.6,72,0\right)$ .

Calculation:

The translation matrix will add 23.6 to each $x$ -coordinate and 72 to each $y$ -coordinate. The $z$ -coordinate of the transformation matrix will remain same, so the resultant matrix will be as follows:

  $\left[\begin{array}{rrrrr}\hfill 23.6& \hfill 23.6& \hfill 23.6& \hfill 23.6& \hfill 23.6\\ \hfill 72& \hfill 72& \hfill 72& \hfill 72& \hfill 72\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]+\left[\begin{array}{rrrrr}\hfill 20& \hfill 136& \hfill 247& \hfill 302& \hfill 351\\ \hfill -58& \hfill -71& \hfill -74& \hfill -83& \hfill -62\\ \hfill 27& \hfill 53& \hfill 59& \hfill 37& \hfill 52\end{array}\right]=\left[\begin{array}{rrrrr}\hfill 43.6& \hfill 159.6& \hfill 270.6& \hfill 325.6& \hfill 374.6\\ \hfill 14& \hfill 1& \hfill -2& \hfill -11& \hfill 10\\ \hfill 27& \hfill 53& \hfill 59& \hfill 37& \hfill 52\end{array}\right]$

c.

To determine

To describe: the result of translation.

Answer to Problem 20E

The matrix has a translation of 23.6 units along $x$ -axis and 72 units along the $y$ -axis.

Explanation of Solution

Given information:

The matrix representing the group of dolphins with the ship at the origin is shown below.

  $\left[\begin{array}{rrrrr}\hfill 20& \hfill 136& \hfill 247& \hfill 302& \hfill 351\\ \hfill -58& \hfill -71& \hfill -74& \hfill -83& \hfill -62\\ \hfill 27& \hfill 53& \hfill 59& \hfill 37& \hfill 52\end{array}\right]$

Later the matrix is translated into a fixed reference point using the vector $\left(23.6,72,0\right)$ .

Calculation:

The result of matrix after transformation is $T\left(x,y,z\right)\to \left(x+23,y+72,z\right)$ i.e. matrix has a translation of 23.6 units along $x$ -axis and 72 units along the $y$ -axis.