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I am currently focused on Chapter 2: Derivation ... ...

I need help with an interpretation of a statement by Kantorovitz near to the start of Chapter 2 ...

The start to Chapter 2 in Kantorovitz reads as folows:

View attachment 7800

https://www.physicsforums.com/attachments/7801

At the end of the above quoted text we read the following:

" ... ... In general, for any unit vector \(\displaystyle u \in \mathbb{R}^k\) and \(\displaystyle x \in \mathbb{R}^k\), the axis through \(\displaystyle x\) in the direction \(\displaystyle u\) is the directed line with the parametric representation

\(\displaystyle \gamma \ : \ t \in \mathbb{R} \rightarrow \gamma (t) := x + tu \in \mathbb{R}^k\) ... ... "

My question is as follows:

Why does Kantorovitz refer to the above line as "the axis through \(\displaystyle x\) in the direction \(\displaystyle u\)" ... surely it is just a line as in my diagram below showing the line through \(\displaystyle x\) in the direction \(\displaystyle u\) in \(\displaystyle \mathbb{R}^3\) ... it is not an axis but simply a line ..https://www.physicsforums.com/attachments/7802The required equation of the line, I think, arises as follows:Consider an arbitrary point, \(\displaystyle P\), on the line given by \(\displaystyle \gamma (t)\) where \(\displaystyle t \in \mathbb{R}\).\(\displaystyle u \in \mathbb{R}^k\) is a vector parallel to the direction of the line ...... we have that \(\displaystyle \gamma (t) = OP\)\(\displaystyle \Longrightarrow \gamma (t) = OP_0 + tu\)\(\displaystyle \Longrightarrow \gamma (t) = x + tu \)

Is that a correct interpretation of the line/axis through x in the direction u ... ?

Peter