Please show all work, step by step. Thank you so much!

Background In computer graphics [4] [as well as in robotics [5]} object positions can be defined using vectors. Thus, changes in position [translations and rotations} can then be dened as matrix operations on these vectors. Likewise, scaling [resizing] the object can be also defined via matrix operations. These operations are represented as, p' = ,0 +T (Translation) (1) p' 2 RP (Rotation) (2) p' = 5P (Scaling) (3) We see that the latter two ofthese operations are multiplications, while the first [translation] is a sum. Thus it is difficult to combine the three operations into a single matrix transformation. However, this combination may be achieved if the vectors are rewritten in homogeneous coordinates. In homogeneous coordinates, we add a third coordinate to a point {vector}. Instead of being represented by a pair of numbers (1,131, each point [vector] is represented as a triple (133/, w). Thus a point in 2D will be represented by the set of points {{x,y, w)|w :t D} and we say point points are equal {equivalent} if and only if one is a multiple of the other. For example, [2,3,6] and [4,6,12} represent the point in 2D. That is, each point has many different homogeneous coordinate representations. Also, at least one coordinate must be nonzero {that is, [0,0,0] is not allowed.} If the w coordinate is nonzero, we can divide through by it giving (1,1, 1). That is, (1,1,1) W W W W represents the same point as (x, y, w) and the numbers% and are called the Cartesian coordinates of the homogeneous point. Hence we generally represent the point (13):) in Cartesian '- In this project, we'll deal with two dimensional vectors for ease ofvisualization, but similar techniques apply to three dimensional vectors [for example, see [2].} In this project, we'll deal with two dimensional vectors for ease of visualization, but similar techniques apply to three dimensional vectors (for example, see [2].) coordinates as the triple (x, y, 1) in homogeneous coordinates. The points with w = 0 are called points at infinity and will not be discussed here. The advantage of homogeneous coordinates is that they allow us to combine the three operations above into a single matrix transformation. For example, the translation of (x, y) by the vector (h, k) (that is, (x, y) is translated to (x, y) + (h, k) = (x + h,y + k)) may be represented as, A counterclockwise rotation through an angle d is given by multiplication by the matrix cos 8 - sin 8 sin e cos 8 0 0 And a scaling by s in the x direction and t in the y direction, is given by, 0 (3') Hence, combinations of these three operations are simply products of matrices of these forms. Note that if the matrix is of the form, 1,1 11,2 12,1 12,2 ty 0 0 where the upper 2 x 2 submatrix is orthogonal and thus preserves angles and lengths (scaling is one in all directions). Such transformations are called rigid-body transformations and are useful in robotics and other applications with no deformations (see [5] for more details).Example For example, the 3 x 3 matrix that corresponds to a scaling of onehalf [0.5), a rotation of 90 degrees, and a translation of l in thexdirection and by 2 in the ydirection, is given by, I] 0.5 1 [0.5 I] 2 l] l] l Here is the transformation applied to a [blue] rectangle in Matlab and the resulting [red] rectangle. Exercises and Explorations 1. [30] Verify that the operations {1}, [2), and [3] are given by the matrix operations as described by [1'], {2'}, and [3']. Verify by comparing the operations and results in Cartesian and homogeneous transformations