Answer to Question #136276 in Astronomy | Astrophysics for Harsh Deep Rautela

Question #136276
Imagine that you are measuring the location of two objects using cylindrical polar coordinates. The first is at r=10.32 , θ=−0.32 and z=45.3 . The second is at r=10.33 , θ=−0.31 and z=45.27 .

How far apart are these two points?
1
Expert's answer
2020-10-07T07:24:25-0400

Solution

We are given the cylindrical coordinates. we have to convert these to regular coordinates of the cartesian plane as shown below;

First point;


to find "x" , we apply "x= r \u00d7 cos(\\theta)"

"= 10.32\u00d7 cos(-0.32)"


"=9.7961"


to find "y" , we apply "y = r\u00d7 sin(\\theta)" `


"= 10.32\u00d7 sin(-0.32)"


"= -3.2463"


And "z" remains the same i.e. "z= 45.3"


Now the first point "p_1 = (9.7961, -32463, 45.3)"


Second point;


Finding "x" ; "x= r \u00d7 cos(\\theta)"


"= 10.33\u00d7 cos (-0.31)"


"= 9.8376"

`

Finding "y" ; "y = r\u00d7 sin(\\theta)"


"= 10.33 \u00d7 sin (-0.31)"


"=-3.1513"


And "z" remains the same i.e. "z= 45.27"


The second point, "p_2 = (9.8376, -3.1513, 45.27)"


Now having the position of the two objects, we find the distance between them; which is given by;


"D^2 = X^2 + Y^2 + Z^2"


To find the "X, Y, and, Z" , we Calculate the positive difference between each coordinates, i.e

"X= 9.8376-9.7961"


"=0.0415"


"Y = -3.1513 - -3.2463"


"=0.095"


"Z = 45.3-45.27"


"=0.03"


Now having the values, we calculate;


"D^2 = X^2 + Y^2 + Z^2"


"= 0.0415^2 + 0.095^2 +0.03^2"


"= 0.00172225 + 0.009025 +0.0009"


"= 0.01164725"


"D = \\sqrt{ 0.01164725}"


"= 0.1079"

Therefore the distance between the two objects is "0.1079" "Units"


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