Question

Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point.

a.\(\left( {\sqrt {\rm{2}} {\rm{,3\pi /4,2}}} \right)\)

b.\(\left( {{\rm{1,1,1}}} \right)\)

Short answer

(a) The point is \(\left( {{\rm{ - 1,1,2}}} \right)\).

(b) The point is \(\left( {{\rm{cos1, sin1,1}}} \right) \approx \left( {{\rm{0}}{\rm{.54,0}}{\rm{.84,1}}} \right)\).

Step-by-step solution

Concept Introduction

A multiple integral is a definite integral of a function of many real variables in mathematics (particularly multivariable calculus), such as f(x, y) or f(x, y) (x, y, z). Integrals of a two-variable function over a region

Step 2:Find the rectangular coordinates of the point \(\left( {\sqrt {\rm{2}} {\rm{,3\pi /4,2}}} \right)\)

(a)The xx axis is represented by the redcolour.

The yy axis is represented by the green colour.

The zz axis is represented by the blue colour.

1)To determine rectangular coordinates, use the formulas below:

\(\begin{aligned}\rm x &= rcos\theta \\\rm y &= rsin\theta \\\rm z &= z\end{aligned}\)

2) Let's figure out our coordinates by solving these equations:

\(\begin{aligned}\rm x &= \sqrt {\rm{2}} {\rm{cos}}\frac{{{\rm{3\pi }}}}{{\rm{4}}}\\\rm &= \sqrt {\rm{2}} *\left( {\frac{{{\rm{ - }}\sqrt {\rm{2}} }}{{\rm{2}}}} \right)\\\rm &= - 1\end{aligned}\)

\(\begin{aligned}\rm y &= \sqrt {\rm{2}} {\rm{sin}}\frac{{{\rm{3\pi }}}}{{\rm{4}}}\\\rm &= \sqrt {\rm{2}} *\left( {\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}} \right)\\\rm &= 1\end{aligned}\)

3) As a result, our answer is:

\(\left( {{\rm{ - 1,1,2}}} \right)\)

Find the rectangular coordinates of the point \(\left( {{\rm{1,1,1}}} \right)\)

(b)

A point \({\rm{P}}\) in three-dimensional space is represented in the cylindrical coordinate system by the ordered triple \(\left( {{\rm{r,\theta ,z}}} \right)\), where \({\rm{r}}\) and \({\rm{\theta }}\) are polar coordinates of the projection of \({\rm{p}}\) onto the \({\rm{xy}}\)-plane, and\({\rm{\;z}}\) is the directed distance from the \({\rm{xy}}\)-plane to \({\rm{p}}\). We utilise the equations to transform from cylindrical to rectangular coordinates.

\(\begin{aligned}\rm x &= rcos\theta ,\;\\\rm y &= rsin\theta ,\;\\\rm z &= z\end{aligned}\)

Now we must determine the provided point's rectangle coordinates.

Here

\(\begin{aligned}\rm r &= 1,\;\\\rm \theta &= 1,\;\\\rm z &= 1\end{aligned}\)

The rectangular coordinates of the object are now

\(\begin{aligned}\rm x &= 1*cos(1)\\\rm &= 0{\rm{.54}}\\\rm y &= 1*sin(1)\\\rm &= 0{\rm{.84}}\\\rm z &= 1{\rm{.}}\end{aligned}\)

In rectangular coordinates, the position is \(\left( {{\rm{0}}{\rm{.54,0}}{\rm{.84,1}}} \right){\rm{.}}\)

In rectangular coordinates, the point is \(\left( {{\rm{cos1, sin1,1}}} \right) \approx \left( {{\rm{0}}{\rm{.54,0}}{\rm{.84,1}}} \right)\).