Question

How do you convert the point $(3, - 3,7)$ from rectangular coordinates to cylindrical coordinates?

Answer 1

Hint: A cylindrical coordinates is a three-dimensional coordinate system that specifies the position of a particular point by the distance from a chosen reference axis, the direction from the axis relative to the chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.
The cylindrical coordinates with respect to the rectangular Cartesian coordinates can be written as:
$r = \sqrt {{x^2} + {y^2}} $, $\phi = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$ and $z = z$. Where $(r, \phi, z)$ are the cylindrical coordinates or respective $(x, y, z)$ rectangular coordinates.

Complete step by step answer:
In the given question we are provided with the point $(3, - 3,7)$ in the rectangular coordinates. We are supposed to convert these given coordinates into a cylindrical coordinate system. To do so, we will first use the formula,
$r = \sqrt {{x^2} + {y^2}} $
Let $x = 3$ and $y = - 3$
On substituting the values of ‘x’, ‘y’, and ‘z’ in the above formula we get,
$ \Rightarrow r = \sqrt {{{(3)}^2} + {{( - 3)}^2}} $
On further simplification we get ,
$
\Rightarrow r = \sqrt {9 + 9} \\
\Rightarrow r = \sqrt {18} \\
\Rightarrow r = 3\sqrt 2 \\
 $
Now let us solve for $\phi $ ,
$
\Rightarrow \phi = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) \\
\Rightarrow \phi = {\tan ^{ - 1}}\left( {\dfrac{{ - 3}}{3}} \right) \\
\Rightarrow \phi = {\tan ^{ - 1}}( - 1) \\
\Rightarrow \phi = - \dfrac{\pi }{4} \\
 $
Hence, Cylindrical coordinates are $\left( {3\sqrt 2, - \dfrac{\pi }{4},7} \right)$.

Note: The three coordinates $(r,\phi ,z)$ of a point P are defined as, the axial distance or radial distance $r$ is the Euclidean distance from z-axis to the point P. The azimuth $\phi $ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane. The axial coordinate or height $z$ is the signed distance from the chosen plane to the point P.