Question

Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. (a) (7,7/3, 7/6) 3 3 لا | من | л л 6 6 o (x, y, z) =

Answer 1

Solution -  

a)   

(7, 3 6

Comparing it with

(r, , છે)

$r=7 ,\theta =\frac{\pi }{3},\phi =\frac{\pi }{6}$

   r is radius or distance of point from origin.$\theta$ is the azimuthal angle in xy plane and  $\phi$ is the polar angle.

So it is graph of point -

  

TH J

$x=rcos\theta sin\phi = 7cos(\frac{\pi }{3})sin(\frac{\pi }{6})$

$= 7(\frac{1}{2})(\frac{1}{2})=\frac{7}{4}$

  $y=rsin\theta sin\phi =7sin(\frac{\pi }{3})sin(\frac{\pi }{6})$

  $=7(\frac{\sqrt{3}}{2})(\frac{1}{2})=\frac{7\sqrt{3}}{4}$

$z=rcos\phi$

$=7 cos(\frac{\pi }{3})=7(\frac{1}{2})= \frac{7}{2}$

So rectangular coordinates of point is $(\frac{7}{4},\frac{7\sqrt{3}}{4},\frac{7}{2})$

b)

$(3,\frac{\pi }{2},\frac{3\pi }{4})$

  $r=3,\theta =\frac{\pi }{2},\phi =\frac{3\pi }{4}$

So graph is following -

  

z 3л 4

$x=rcos\theta sin\phi = 3cos(\frac{\pi }{2})sin(\frac{3\pi }{4})$

$=0$

$y=rsin\theta sin\phi =3sin(\frac{\pi }{2})sin(\frac{3\pi }{4})$

$=\frac{3\sqrt{2}}{2}$

$z=rcos\phi =7cos\left ( \frac{3\pi }{4} \right )=-\frac{3\sqrt{2}}{2}$

So rectangular coordinates of point is $(0,\frac{3\sqrt{2}}{2},-\frac{3\sqrt{2}}{4})$ Answer