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×cos(θ)
=10.32×cos(−0.32)
=9.7961
to find y , we apply y=r×sin(θ) `
=10.32×sin(−0.32)
=−3.2463
And z remains the same i.e. z=45.3
Now the first pointp1=(9.7961,−32463,45.3)
Second point;
Finding x ; x=r×cos(θ)
=10.33×cos(−0.31)
=9.8376
`
Finding y ; y=r×sin(θ)
=10.33×sin(−0.31)
=−3.1513
And z remains the same i.e. z=45.27
The second point, p2=(9.8376,−3.1513,45.27)
Now having the position of the two objects, we find the distance between them; which is given by;
D2=X2+Y2+Z2
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;
D2=X2+Y2+Z2
=0.04152+0.0952+0.032
=0.00172225+0.009025+0.0009
=0.01164725
D=0.01164725
=0.1079
Therefore the distance between the two objects is 0.1079Units
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