Can the Dot Product Determine Maximum Distance from the Origin?

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Homework Help Overview

The discussion revolves around the use of the dot product of a vector with itself to determine maximum distance from the origin, specifically exploring the relationship between the dot product, derivatives, and distance in a vector context.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to use the dot product of a vector with itself to find maximum distance and questions the validity of their approach based on graphical results. Some participants discuss the implications of the derivative of the distance squared being zero and its relation to perpendicular vectors.

Discussion Status

Participants are exploring the mathematical reasoning behind the conditions for maximum distance, with some guidance provided regarding the relationship between the position and velocity vectors. There is an ongoing inquiry into the reasoning behind the conditions discussed, indicating a productive exploration of the concepts.

Contextual Notes

There is a mention of a graph created using Wolfram Alpha that contradicts the original poster's findings, suggesting potential discrepancies in understanding or application of the concepts involved.

jimmy42
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Hello,

If I have a vector A and then I do the dot product on itself so A°A. Then can I use that to find the maximum distance from the origin? If I take the derivative of the dot product then can I know at what time the maximum distance was travelled?

I have done this but it is wrong based on the graph I made using Wolfram Alpha, I just need some reassurance that I'm on the right track.
 
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hello jimmy42! :smile:

your method looks ok …

distance2 = A.A,

so if the distance is a maximum, then A'.A = 0

ie A' is perpendicular to A
 
Why if A'.A =0 is the distance a maximum?

I too have a question on this, and I'm failing to see why if the position vector and the velocity vector are perpendicular then the distance is a maximum if the above holds true.

Cheers in advance

Smithy
 
Welcome to PF!

Hi Smithy! Welcome to PF! :smile:
smith873 said:
Why if A'.A =0 is the distance a maximum?

Because that's the derivative of the distance squared (divided by 2),

so it must be 0 if the distance squared is at a turning-point (maximum minimum or inflection point). :wink:
 

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