(c) Determine the concatenated transformation matrix for translation by vector [1 1] followed by rotation of 45 degrees in 2D as shown in Figure 2 below.

To solve this problem, we need to understand that transformations in 2D space can be represented by matrices.

Step 1: Translation Matrix The translation matrix for a 2D space is given by:

[1 0 Tx]
[0 1 Ty]
[0 0 1 ]

Where Tx and Ty are the translation distances along the x and y axes respectively. In this case, both Tx and Ty are 1. So, the translation matrix is:

[1 0 1]
[0 1 1]
[0 0 1]

Step 2: Rotation Matrix The rotation matrix for a 2D space is given by:

[cosθ -sinθ 0]
[sinθ  cosθ 0]
[ 0     0   1]

Where θ is the angle of rotation. In this case, θ is 45 degrees. So, the rotation matrix is:

[cos(45) -sin(45) 0]
[sin(45)  cos(45) 0]
[  0        0     1]

Step 3: Concatenated Transformation Matrix The concatenated transformation matrix is obtained by multiplying the translation matrix and the rotation matrix. The order of multiplication is important. Since the translation is followed by the rotation, we first multiply the rotation matrix with the translation matrix.

The final concatenated transformation matrix is the result of this multiplication.