Question on Vectors and Components (NOT HW)

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Homework Help Overview

The discussion revolves around the concepts of vector projections and components in the context of vector mathematics. The original poster seeks clarification on the differences between the projection of a vector onto another and the components of that vector along specified directions.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the definitions and geometric interpretations of vector projections and components. There is a discussion on how projections involve dropping perpendiculars and measuring distances, while components relate to expressing a vector as a linear combination of others. Some participants question whether these concepts are visually equivalent.

Discussion Status

The conversation is ongoing, with participants providing insights into the mathematical definitions and geometric interpretations of the concepts. There is an acknowledgment of differing perspectives on the visual representation of projections and components, but no consensus has been reached.

Contextual Notes

Participants are navigating the nuances of vector mathematics, particularly in relation to inner product spaces and the implications of orthogonality on projections and components.

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Okay, let's say there's vector a,b and c

What would be the difference between


projection of c on a and b

and

components of c along a and b


I thought they were the same...

thanks.
 
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The projections are like putting c and a or b tail to tail and dropping a perpendicular from the tip of c to the line defined by a or b and measuring from the tail of a or b to the intersection. In an inner product space with inner product <.,.>, the projection onto b can be calculated by <b, c>/||b||.
The components of c with respect to a and b are the numbers ca and cb such that c = caa + cbb. In the case that c is the sum of a and b, the components with respect to a and b are (1, 1). By the parallelogram law, this obviously does not correspond to the projection unless a and b are orthogonal unit vectors.
 
hmm... but drawing wise, they are the same?
 
Drawing wise they should be the same as saying what the poster above said by
grabbing the vector a and the vector c (we are projecting a onto c in this case) and its just the length of a in the c direction.

With regards to your question I'm pretty sure you have the right idea.
 

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