Transformation example

Here’s the step-by-step breakdown of combining the transformations and applying them to the point $P$:

Combining Scaling and Rotation

First, we multiply the rotation and scaling matrices:

$$\text{Rotate\_Scale}=\text{Rotate}\times \text{Scale}=\left[\begin{array}{ccc}\sqrt{2}/2& -\sqrt{2}/2& 0\\ \sqrt{2}/2& \sqrt{2}/2& 0\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1.414& -1.414& 0\\ 1.414& 1.414& 0\\ 0& 0& 1\end{array}\right]$$

Adding Translation

Next, we incorporate translation by multiplying the result with the translation matrix:

$$\text{Combined}=\text{Translate}\times \text{Rotate\_Scale}=\left[\begin{array}{ccc}1& 0& 5\\ 0& 1& 3\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{ccc}1.414& -1.414& 0\\ 1.414& 1.414& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1.414& -1.414& 5\\ 1.414& 1.414& 3\\ 0& 0& 1\end{array}\right]$$

Applying to a Point (P)

Lastly, applying this combined matrix to a point $P=[1,1,1]$ gives us the new position:

$$P^{\prime }=\text{Combined}\times P=\left[\begin{array}{ccc}1.414& -1.414& 5\\ 1.414& 1.414& 3\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 5.828\\ 1\end{array}\right]$$

Thus, the original point $P(1,1)$ is transformed to $P^{\prime }(5,5.828)$ after applying the scaling, rotation, and translation in one combined step.