Question

How do you change (0,3,-3) from rectangular to spherical coordinates?

Answer 1

Hint: Spherical coordinates are also called spherical polar coordinates. The spherical polar coordinate system is denoted as $\left( {r,\theta ,\Phi } \right)$ which is mainly used in three dimensional systems. Through these coordinates, three numbers are specified : the radial distance, the polar angles, and the azimuthal angle. Also, these coordinates are determined with the help of Cartesian coordinates(x,y,z). The radius can be measured from a fixed origin, the polar angle can be measured from an imaginary point which is near to the exact location, and azimuthal angle passed through the origin on the reference plane. ‘r’ is the radius of the system, $\theta $ is an inclination angle and $\Phi $ is azimuth angle.
The spherical coordinates with respect to the Cartesian coordinates can be written as:
$r = \sqrt {{x^2} + {y^2} + {z^2}} $
$\tan \theta = \dfrac{{\sqrt {{x^2} + {y^2}} }}{z}$
$\tan \Phi = \dfrac{y}{z}$

Complete step by step solution:
In this question, we want to convert (0,3,-3) from rectangular to spherical coordinates.
From the given coordinate, the value of x is 0, the value of y is 3 and the value of z is -3.
First, let us find the radius r.
$ \Rightarrow r = \sqrt {{x^2} + {y^2} + {z^2}} $
Let us substitute all the values.
$ \Rightarrow r = \sqrt {{{\left( 0 \right)}^2} + {{\left( 3 \right)}^2} + {{\left( { - 3} \right)}^2}} $
That is equal to,
$ \Rightarrow r = \sqrt {9 + 9} $
Apply addition on the right-hand side.
$ \Rightarrow r = \sqrt {18} $
Therefore,
$ \Rightarrow r = 3\sqrt 2 $
Now, let us find the value of the polar angle.
$ \Rightarrow \tan \theta = \dfrac{{\sqrt {{x^2} + {y^2}} }}{z}$
Let us substitute all the values.
$ \Rightarrow \tan \theta = \dfrac{{\sqrt {{{\left( 0 \right)}^2} + {{\left( 3 \right)}^2}} }}{{ - 3}}$
That is equal to,
$ \Rightarrow \tan \theta = \dfrac{{\sqrt 9 }}{{ - 3}}$
Apply square-root on the right-hand side.
$ \Rightarrow \tan \theta = \dfrac{3}{{ - 3}}$
Therefore,
$ \Rightarrow \tan \theta = - 1$
$ \Rightarrow \theta = \arctan \left( { - 1} \right)$
That is equal to,
$ \Rightarrow \theta = \dfrac{{3\pi }}{4}$
Now, let us find the value of the azimuth angle.
$ \Rightarrow \tan \Phi = \dfrac{y}{z}$
Let us substitute all the values.
$ \Rightarrow \tan \Phi = \dfrac{{ - 3}}{0}$
That is equal to,
$ \Rightarrow \tan \Phi = \infty $
Therefore,
$ \Rightarrow \Phi = \arctan \left( \infty \right)$
That is equal to,
$ \Rightarrow \Phi = \dfrac{\pi }{2}$

Hence, the coordinate is $\left( {3\sqrt 2 ,\dfrac{{3\pi }}{4},\dfrac{\pi }{2}} \right)$.

Note:
In three-dimensional space, the spherical polar coordinate system is used for finding the surface area. We can also call the radial distance a radial coordinate apart from the radius. The polar angle can be mentioned as colatitudes, zenith angle, normal angle, or inclination angle.