MHB How Do You Calculate the Minimal Path Sum in Coordinate Geometry?

  • Thread starter Thread starter Albert1
  • Start date Start date
AI Thread Summary
The minimal path sum in coordinate geometry involves calculating the shortest distance between points P=(3,7) and Q=(1,5) via points A and B. By reflecting point Q across the y-axis and then the x-axis, the shortest distance is determined to be the straight line to the reflected point Q''. The length of the path, denoted as k, equals 4√10. The coordinates for points A and B are found to be A=(2/3,0) and B=(0,2). The final result for x+y+k is 4√10 + 8/3.
Albert1
Messages
1,221
Reaction score
0
P=(3,7),Q=(1,5) ,A=(x,0),B=(0,y)

if k=minimum(PA+AB+BQ)

find: x+y+k
 
Mathematics news on Phys.org
[sp]

Let $Q'$ be the reflection of $Q$ in the $y$-axis and let $Q''$ be the reflection of $Q'$ in the $x$-axis. The shortest distance between $P$ and $Q''$ is the straight line between them. The shortest path between $P$ and $Q$ (via $A$ and $B$) is obtained by reflecting parts of that line in the axes, as appropriate (the red path in the diagram). The length $k$ of the path is the same as the distance from $P$ to $Q''$, which is $4\sqrt{10}.$ The point $A$ is at $(2/3,0)$ and $B$ is at $(0,2)$. So $x+y+k = 4\sqrt{10} + \frac83$.[/sp]
 

Attachments

  • reflections.png
    reflections.png
    4.4 KB · Views: 68
Opalg said:
[sp]
https://www.physicsforums.com/attachments/2267​

Let $Q'$ be the reflection of $Q$ in the $y$-axis and let $Q''$ be the reflection of $Q'$ in the $x$-axis. The shortest distance between $P$ and $Q''$ is the straight line between them. The shortest path between $P$ and $Q$ (via $A$ and $B$) is obtained by reflecting parts of that line in the axes, as appropriate (the red path in the diagram). The length $k$ of the path is the same as the distance from $P$ to $Q''$, which is $4\sqrt{10}.$ The point $A$ is at $(2/3,0)$ and $B$ is at $(0,2)$. So $x+y+k = 4\sqrt{10} + \frac83$.[/sp]
your answer is correct:)
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top