Question

If $x=r\sin \theta \cdot \cos \varphi ,$ $y=r\sin \theta \cdot \sin \varphi$ and $z=r\cos \theta$, then the value of $x^2+y^2+z^2$ is independent of:

• A. $\theta ,\varphi$
• B. $r,\theta$
• C. $r,\varphi$
• D. $r$

Answer 1

Answer: (A)

$\theta ,\varphi$

Given, $x=r\sin \theta \cdot \cos \varphi ,$ $y=r\sin \theta \cdot \sin \varphi$ and $z=r\cos \theta$

Therefore, $x^2+y^2+z^2$

$={\left(r\sin \theta \cos \varphi \right)}^2+{\left(r\sin \theta \sin \varphi \right)}^2+{\left(r\cos \theta \right)}^2$

$=r^2{\sin }^2\theta {\cos }^2\varphi +r^2{\sin }^2\theta {\sin }^2\varphi +r^2{\cos }^2\theta$

$=r^2{\sin }^2\theta ({\cos }^2\varphi +{\sin }^2\varphi )+r^2{\cos }^2\theta$

$=r^2{\sin }^2\theta +r^2{\cos }^2\theta$

$=r^2({\sin }^2\theta +{\cos }^2\theta )$

$=r^2$