Translation from object space to the world space

I’m stuyding computer animation with this book, Computer animation by The Morgan Kaufmann series. The explanation for translation from object space to world space is difficult to understand. In more detail, what I thought is not same to the textbook.

I have read this article, but I’m not sure is it helpful to solve this problem. How to convert from Object space into World space (exercise from 3D Math Primer book)

This is the way that I understood vs the textbook explains.

The difference between two of them is what is the orientation vector in z? The book explained that the endpoint of the orientation vector is at 0,0,sqrt(50). But why suddenly sqrt(34) came up?.. I don’t know why the the endpoint should be in 0,0,sqrt(50). Is it would be in 5? It is so confused to me.

[Textbook explains]

• You drew the picture wrong. When the vector $(0,0, \sqrt50)$ is rotated so it's $y$ coordinate is now $-4$ the Z value would automatically decrease. You subconsciously elongated that vector to think it's Z value is still lying at $\sqrt50$. Think of it like this, are the vector $(1,0)$ and vector $(1,1)$ of the same length? The answer is no, if you can figure that out using the pythagorean rule you will get your answer :) Oct 21, 2022 at 5:25