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I am reading David Bachman's book: "A Geometric Approach to Differential Forms" (Second Edition) ...

I need some help with some remarks Bachman makes near the start of his section on 1-forms ...

The relevant section reads as follows:View attachment 8604In the above text from Bachman we read the following:

" ... ... Recall the geometric interpretation of the dot product: You project \(\displaystyle \langle -1, 2 \rangle\) onto \(\displaystyle \langle 2, 3 \rangle\) and then multiply by \(\displaystyle \mid \langle 2, 3 \rangle \mid = \sqrt{13}\). ... ... "My question is as follows:

Is there any reason Bachman has chosen the projection of \(\displaystyle \langle -1, 2 \rangle\) onto \(\displaystyle \langle 2, 3 \rangle\) ... ... ?

Could he just have easily chosen the projection of \(\displaystyle \langle 2, 3 \rangle\) onto \(\displaystyle \langle -1, 2 \rangle\) ... ... ?Can someone please clarify ...

Peter==================================================================================

It may help MHB readers of the above post to have access to Bachman's Section 3.1 ... ... so I am providing the same .. ... as follows:View attachment 8605

View attachment 8606Hope that helps ...

Peter

I need some help with some remarks Bachman makes near the start of his section on 1-forms ...

The relevant section reads as follows:View attachment 8604In the above text from Bachman we read the following:

" ... ... Recall the geometric interpretation of the dot product: You project \(\displaystyle \langle -1, 2 \rangle\) onto \(\displaystyle \langle 2, 3 \rangle\) and then multiply by \(\displaystyle \mid \langle 2, 3 \rangle \mid = \sqrt{13}\). ... ... "My question is as follows:

Is there any reason Bachman has chosen the projection of \(\displaystyle \langle -1, 2 \rangle\) onto \(\displaystyle \langle 2, 3 \rangle\) ... ... ?

Could he just have easily chosen the projection of \(\displaystyle \langle 2, 3 \rangle\) onto \(\displaystyle \langle -1, 2 \rangle\) ... ... ?Can someone please clarify ...

Peter==================================================================================

It may help MHB readers of the above post to have access to Bachman's Section 3.1 ... ... so I am providing the same .. ... as follows:View attachment 8605

View attachment 8606Hope that helps ...

Peter