Finding the distance between two polar coordinates
Two options for finding the distance between polar points
To find the distance between two polar coordinates, we have two options.
We can either convert the polar points to rectangular points, then use a simpler distance formula, or we can skip the conversion to rectangular coordinates, but use a more complicated distance formula.
Here’s a summary of our options:
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Examples for finding the distance between polar points using both methods
Example
Find the distance between the polar points.
???\left(2,\frac{\pi}{2}\right)??? and ???\left(3,\frac{\pi}{4}\right)???
We’ll convert the polar points into cartesian points.
Using the conversion formulas to change ???\left(2,\frac{\pi}{2}\right)??? into polar, we get
???x_1=2\cos{\frac{\pi}{2}}???
???x_1=0???
and
???y_1=2\sin{\frac{\pi}{2}}???
???y_1=2???
The new point is ???(0,2)???.
Using the conversion formulas to change ???\left(3,\frac{\pi}{4}\right)??? into polar, we get
???x_2=3\cos{\frac{\pi}{4}}???
???x_2=\frac{3\sqrt{2}}{2}???
and
???y_2=3\sin{\frac{\pi}{4}}???
???y_2=\frac{3\sqrt{2}}{2}???
The new point is ???\left(\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right)???.
Setting
???(x_1,y_1)=(0,2)???
???(x_2,y_2)=\left(\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right)???
and plugging these points into the distance formula from the cartesian coordinate system, we get
???D=\sqrt{\left(\frac{3\sqrt{2}}{2}-0\right)^2+\left(\frac{3\sqrt{2}}{2}-2\right)^2}???
???D=\sqrt{\frac{18}{4}+\frac{18}{4}-\frac{12\sqrt{2}}{2}+4}???
???D=\sqrt{13-6\sqrt{2}}???
The distance between the polar points is ???D=\sqrt{13-6\sqrt{2}}???.
Let’s look at a different example where we use the second method.
We can either convert the polar points to rectangular points, then use a simpler distance formula, or we can skip the conversion to rectangular coordinates, but use a more complicated distance formula.
Example
Find the distance between the polar points.
???(1,2\pi)??? and ???(2,\pi)???
Setting
???(r_1,\theta_1)=(1,2\pi)???
???(r_2,\theta_2)=(2,\pi)???
and plugging these points into the distance formula from the polar coordinate system, we get
???D=\sqrt{(1)^2+(2)^2-2(1)(2)\cos{(2\pi-\pi)}}???
???D=\sqrt{5-4\cos{\pi}}???
???D=\sqrt{5-4(-1)}???
???D=\sqrt{5+4}???
???D=\sqrt{9}???
???D=3???
The distance between the polar points is ???D=3???.
