Chapter 3.5, Problem 30PPS

(a)

To determine

To write:A matrix to represent a given triangle.

Answer to Problem 30PPS

The matrix to represent the given triangle is $\left[\begin{array}{ccc}4& -1& -3\\ 2& -1& 1\end{array}\right]$

Explanation of Solution

Given information:

The graphical representation of triangle ABC

Formula used:

1) The coordinates of the vertices of the triangle is identified in the form of ordered pair.

2) Each ordered pair is represented as the column of a matrix to get a 3×2 matrix.

Calculation:

Consider the figure given below. Assuming each division as 1 unit, the coordinates of the vertices are obtained.

  

The vertices as obtained from the figure are $A(4,2),B(-1,-1),C(-3,1)$ .

Now, representing each ordered pair of coordinates as column of a matrix, the triangle $\Delta ABC$ can be represented in matrix form as,

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

b.

To determine

To represent: The given matrix geometrically

Explanation of Solution

Given information:

The matrix representing a given triangle ABC

The matrix representing the triangle ABCas obtained in the previous part is

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

By multiplying the above matrix by 2, the resulting matrix is given by,

  $\begin{array}{c}A\text{'}B\text{'}C\text{'}=2\left[\begin{array}{ccc}4& -1& -3\\ 2& -1& 1\end{array}\right]\\ =\left[\begin{array}{ccc}8& -2& -6\\ 4& -2& 2\end{array}\right]\end{array}$

Now, since each column represents the coordinates of individual vertices of the triangle, the geometric representation can be obtained as follows.

  

c.

To determine

To compare: the figures representing triangles ABC and A’B’C’ . Also to verify the result of multiplying the matrix representing triangle ABC by a different factor.

Explanation of Solution

Given information:

Two triangles ABC and A’B’C’ with their geometric and matrix representation.

The geometric representations of the triangles ABC and A’B’C’as obtained above is

  

By comparing the two graphs it can be observed that, the triangle has been expanded in position by a factor of 2. In terms of the coordinates, the vertices has moved to twice the value of their original position.

Now, when the matrix representing the triangle ABC is multiplied by 0.5, this shall result in the compression of the figure in terms of their positions. The position of the vertices moves to new points which are half their original coordinate value. This can be verified using matrix as well as geometric form as follows.

  

Analytically, this can be verified as,

  $\begin{array}{c}A\text{'}\text{'}B\text{'}\text{'}C\text{'}\text{'}=0.5\left[\begin{array}{ccc}4& -1& -3\\ 2& -1& 1\end{array}\right]\\ =\left[\begin{array}{ccc}2& -0.5& -1.5\\ 1& -0.5& 0.5\end{array}\right]\end{array}$

This agrees with the assumption regarding compression of the size of the triangle.