# Current Affairs JEE Main & Advanced

## Co-ordinates of a Point in Space

Category : JEE Main & Advanced

(1) Cartesian co-ordinates : Let $O$ be a fixed point, known as origin and let $OX,OY$ and $OZ$be three mutually perpendicular lines, taken as x-axis, y-axis and z-axis respectively, in such a way that they form a right-handed system.

The planes $XOY,YOZ$ and $ZOX$are known as xy-plane,  yz-plane and zx-plane respectively.

Also,$OA=x,\,\,OB=y,\,\,OC=z$.

The three co-ordinate planes ($XOY,YOZ$ and$ZOX$) divide space into eight parts and these parts are called octants.

Sign of co-ordinates of a point : The signs of the co-ordinates of a point in three dimension follow the convention that all distances measured along or parallel to $OX,\,\,OY,\,\,OZ$ will be positive and distances moved along or parallel to $OX',\,\,OY',\,\,OZ'$ will be negative.

(2) Cylindrical co-ordinates : If the rectangular cartesian co-ordinates of $P$ are $(x,y,z),$ then those of $N$ are $(x,y,\text{ }0)$ and we can easily have the following relations : $x=u\cos \,\phi ,\,\,y=u\sin \phi$ and $z=z$.

Hence, ${{u}^{2}}={{x}^{2}}+{{y}^{2}}$ and $\varphi ={{\tan }^{-1}}(y/x)$.

Cylindrical co-ordinates of $P\equiv (u,\phi ,z)$

(3) Spherical polar co-ordinates : The measures of quantities $r,\,\,\theta ,\,\,\phi$ are known as spherical or three dimensional polar co-ordinates of the point $P$. If the rectangular cartesian co-ordinates of $P$ are $(x,y,z)$ then $z=r\cos \,\theta ,\,\,u=r\sin \,\theta$.

$\therefore$ $x=u\cos \,\phi =r\sin \,\theta \,\cos \,\phi ,\,\,y=u\sin \,\phi =r\,\sin \theta \,\sin \,\phi$ and $z=r\cos \,\theta$

Also, ${{r}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}$and $\tan \theta =\frac{u}{z}=\frac{\sqrt{{{x}^{2}}+{{y}^{2}}}}{z};\,\,\tan \phi =\frac{y}{x}$.

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