I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 12.13 : Spherical Coordinates
2. Convert the Cartesian coordinates for \(\left( { - 2, - 1, - 7} \right)\) into Spherical coordinates.
Show All Steps Hide All Steps
Start SolutionFrom the point we’re given we have,
\[x = - 2\hspace{0.5in}y = - 1\hspace{0.5in}z = - 7\] Show Step 2Let’s first determine \(\rho \).
\[\rho = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( { - 7} \right)}^2}} = \sqrt {54} \] Show Step 3We can now determine \(\varphi \).
\[\cos \varphi = \frac{z}{\rho } = \frac{{ - 7}}{{\sqrt {54} }}\hspace{0.5in}\varphi = {\cos ^{ - 1}}\left( {\frac{{ - 7}}{{\sqrt {54} }}} \right) = 2.8324\] Show Step 4Let’s use the \(y\) conversion formula to determine \(\theta \).
\[\sin \theta = \frac{{ - 1}}{{\rho \sin \varphi }} = \frac{{ - 1}}{{\sqrt {54} \sin \left( {2.8324} \right)}} = - 0.4472\hspace{0.25in} \to \hspace{0.25in}{\theta _1} = {\sin ^{ - 1}}\left( { - 0.4472} \right) = - 0.4636\]This angle is in the fourth quadrant and if we sketch a quick unit circle we see that a second angle in the third quadrant is \({\theta _2} = \pi + 0.4636 = 3.6052\).
If we look at the three dimensional coordinate system from above we can see that from our \(x\) and \(y\) coordinates the point is in the third quadrant. This in turn means that we need to use \({\theta _2}\) for our point.
The Spherical coordinates are then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\sqrt {54} ,3.6052,2.8324} \right)}}\]