Can the Dot Product Determine Maximum Distance from the Origin?

[jimmy42]

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.

 

[tiny-tim]

hello jimmy42!

your method looks ok …

distance² = A.A,

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

ie A' is perpendicular to A

 

[smith873]

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

 

[tiny-tim]

Welcome to PF!

Hi Smithy! Welcome to PF!

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).

 

[asteriatic]

It seems like you are trying to use the dot product of a vector with itself to find the maximum distance from the origin. However, the dot product only gives you a scalar value, not a distance. To find the maximum distance from the origin, you would need to use the magnitude of the vector, which is the square root of the dot product of the vector with itself.

Taking the derivative of the dot product may give you the time at which the maximum distance was traveled, but it would depend on the specific context of your problem. It's important to keep in mind that the dot product and its derivative are not directly related to distance and time, so it's important to carefully consider the physical meaning behind your calculations.

I would recommend double checking your calculations and making sure you are using the appropriate formulas for finding distance and time. It's always a good idea to have someone else review your work as well for additional reassurance. Best of luck with your research!