# Current Affairs JEE Main & Advanced

#### Projection

Projection of a line joining the points$\mathbf{P(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{z}}_{\mathbf{1}}}\mathbf{)}$ and $\mathbf{Q(}{{\mathbf{x}}_{\mathbf{2}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{2}}}\mathbf{,}{{\mathbf{z}}_{\mathbf{2}}}\mathbf{)}$ on another line whose direction cosines are $\mathbf{l,}\,\,\mathbf{m}$ and $\mathbf{n}$ : Let PQ be a line segment where $P\equiv ({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$ and $Q\equiv ({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ and AB be a given line with d.c.’s as $l,\,\,m,\,\,n$. If the line segment PQ makes angle $\theta$  with the line AB, then       Projection of PQ is $P'Q'=PQ\cos \,\theta$   $=({{x}_{2}}-{{x}_{1}})\cos \alpha +({{y}_{2}}-{{y}_{1}})\cos \beta +({{z}_{2}}-{{z}_{1}})\cos \gamma$   $=({{x}_{2}}-{{x}_{1}})l+({{y}_{2}}-{{y}_{1}})m+({{z}_{2}}-{{z}_{1}})n$.   For x-axis,$l=1,\,\,m=0,\,\,n=0$.    Hence, projection of PQ on x-axis $={{x}_{2}}-{{x}_{1}}$.     Similarly, projection of PQ on y-axis $={{y}_{2}}-{{y}_{1}}$ and projection of PQ on  z-axis $={{z}_{2}}{{z}_{1}}$.

#### Direction Cosines and Direction Ratios

(1) Direction cosines : If $\alpha ,\,\,\beta ,\,\,\gamma$ be the angles which a given directed line makes with the positive direction of the $x,\,\,y,\,\,z$ co-ordinate axes respectively, then $\cos \alpha ,\,\cos \beta ,\,\cos \gamma$  are called the direction cosines of the given line and are generally denoted by $l,\,m,\,n$ respectively.     Thus,  $l=\cos \alpha ,\,\,m=\cos \beta$ and $n=\cos \gamma ,\,\,{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1$.     By definition, it follows that the direction cosine of the axis of $x$ are respectively $\cos {{0}^{o}},\,\,\cos {{90}^{o}},\,\,\cos {{90}^{o}}$ i.e., $(1,\,\,0,\,\,0)$. Similarly direction cosines of the axes of $y$ and $z$ are respectively $(0,\,\,1,\,\,0)$ and $(0,\,\,0,\,\,1)$.     (2) Direction ratios: If $a,b,c$ are three numbers proportional to direction cosines $l,\,\,m,\,\,n$ of a line, then $a,\,\,b,\,\,\,c$ are called its direction ratios. They are also called direction numbers or direction components.     Hence by definition,     $l=\pm \frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$,$m=\pm \frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$,$n=\pm \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$     where the sign should be taken all positive or all negative.     Direction ratios are not unique, whereas d.c.’s are unique.     i.e., ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ne 1$.     (3) D.c.’s and d.r.’s of a line joining two points : The direction ratios of line PQ joining $P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$ and $Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ are ${{x}_{2}}-{{x}_{1}}=a$, ${{y}_{2}}-{{y}_{1}}=b$ and ${{z}_{2}}-{{z}_{1}}=c$, (say).     Then direction cosines are,      $l=\frac{({{x}_{2}}-{{x}_{1}})}{\sqrt{\sum {{({{x}_{2}}-{{x}_{1}})}^{2}}}},\,\text{ }m=\frac{({{y}_{2}}-{{y}_{1}})}{\sqrt{\sum {{({{x}_{2}}-{{x}_{1}})}^{2}}}},\,\text{ }n=\frac{({{z}_{2}}-{{z}_{1}})}{\sqrt{\sum {{({{x}_{2}}-{{x}_{1}})}^{2}}}}$     i.e., $l=\frac{{{x}_{2}}-{{x}_{1}}}{PQ},\,m=\frac{{{y}_{2}}-{{y}_{1}}}{PQ},\,n=\frac{{{z}_{2}}-{{z}_{1}}}{PQ}$.

#### Triangle and Tetrahedron

(1) Co-ordinates of the centroid     (i) If $({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\,({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ and $({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})$ are the vertices of a triangle, then co-ordinates of its centroid are $\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3},\,\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3} \right)$     (ii) If $({{x}_{r}},\,{{y}_{r}},\,{{z}_{r}})$; $r=\text{ }1,\text{ }2,\text{ }3,\text{ }4,$ are vertices of a tetrahedron, then co-ordinates of its centroid are $\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{4},\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}}{4},\,\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}+{{z}_{4}}}{4} \right)$     (2) Area of triangle : Let $A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$, $B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ and $C({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})$ be the vertices of a triangle, then ${{\Delta }_{x}}=\frac{1}{2}\left| \,\begin{matrix} {{y}_{1}} & {{z}_{1}} & 1 \\ {{y}_{2}} & {{z}_{2}} & 1 \\ {{y}_{3}} & {{z}_{3}} & 1 \\ \end{matrix}\, \right|$, ${{\Delta }_{y}}=\frac{1}{2}\left| \,\begin{matrix} {{x}_{1}} & {{z}_{1}} & 1 \\ {{x}_{2}} & {{z}_{2}} & 1 \\ {{x}_{3}} & {{z}_{3}} & 1 \\ \end{matrix}\, \right|$, ${{\Delta }_{z}}=\frac{1}{2}\left| \,\begin{matrix} {{x}_{1}} & {{y}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & 1 \\ {{x}_{3}} & {{y}_{3}} & 1 \\ \end{matrix}\, \right|$     Now, area of $\Delta ABC$ is given by the relation $\Delta =\sqrt{\Delta _{x}^{2}+\Delta _{y}^{2}+\Delta _{z}^{2}}$.   (3) Condition of collinearity: Points $A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),$ $B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ and $C({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})$ are collinear,     If  $\frac{{{x}_{1}}-{{x}_{2}}}{{{x}_{2}}-{{x}_{3}}}=\frac{{{y}_{1}}-{{y}_{2}}}{{{y}_{2}}-{{y}_{3}}}=\frac{{{z}_{1}}-{{z}_{2}}}{{{z}_{2}}-{{z}_{3}}}$.     (4) Volume of tetrahedron : If vertices of tetrahedron be  $({{x}_{r}},\,{{y}_{r}},\,{{z}_{r}})$; $r=\text{ }1,\text{ }2,\text{ }3,\text{ }4;$ then $V=\frac{1}{6}\left| \,\begin{matrix} {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1 \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1 \\ {{x}_{4}} & {{y}_{4}} & {{z}_{4}} & 1 \\ \end{matrix}\, \right|$.

#### Section Formulae

If $P(x,y)$ divides the join of $A({{x}_{1}},{{y}_{1}})$ and $B({{x}_{2}},{{y}_{2}})$ in the ratio ${{m}_{1}}:{{m}_{2}}({{m}_{1}},{{m}_{2}}>0)$             (1) Internal division :  If $P(x,y)$ divides the segment AB internally in the ratio of ${{m}_{1}}:{{m}_{2}}$$\Rightarrow$$\frac{PA}{PB}=\frac{{{m}_{1}}}{{{m}_{2}}}$     The co-ordinates of  $P(x,y)$ are     $x=\frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}$ and $y=\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}$     (2) External division :  If $P(x,y)$divides the segment AB externally in the ratio of ${{m}_{1}}:{{m}_{2}}$$\Rightarrow$$\frac{PA}{PB}=\frac{{{m}_{1}}}{{{m}_{2}}}$           The co-ordinates of $P(x,y)$ are       $x=\frac{{{m}_{1}}{{x}_{2}}-{{m}_{2}}{{x}_{1}}}{{{m}_{1}}-{{m}_{2}}}$ and $y=\frac{{{m}_{1}}{{y}_{2}}-{{m}_{2}}{{y}_{1}}}{{{m}_{1}}-{{m}_{2}}}$

#### Section Formula

(1) Section formula for internal or external division : Let $P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$ and $Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ be two points. Let $R$ be a point on the line segment joining $P$ and $Q$ such that it divides the join of $P$ and $Q$ internally or externally in the ratio ${{m}_{1}}:{{m}_{2}}$.     Then the co-ordinates of $R$ are     $\left( \frac{{{m}_{1}}{{x}_{2}}\pm {{m}_{2}}{{x}_{1}}}{{{m}_{1}}\pm {{m}_{2}}},\,\frac{{{m}_{1}}{{y}_{2}}\pm {{m}_{2}}{{y}_{1}}}{{{m}_{1}}\pm {{m}_{2}}},\,\frac{{{m}_{1}}{{z}_{2}}\pm {{m}_{2}}{{z}_{1}}}{{{m}_{1}}\pm {{m}_{2}}} \right)$.     (2) Co-ordinates of the general point : The co-ordinates of any point lying on the line joining points $P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$ and $Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ may be taken as $\left( \frac{k{{x}_{2}}+{{x}_{1}}}{k+1},\,\frac{k{{y}_{2}}+{{y}_{1}}}{k+1},\,\frac{k{{z}_{2}}+{{z}_{1}}}{k+1} \right)$, which divides $PQ$ in the ratio $k:1$. This is called general point on the line $PQ$.

#### Distance Formula

(1) Distance formula: The distance between two points $A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$ and $B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ is given by   $AB=\sqrt{[{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}]}$.   (2) Distance from origin : Let $O$ be the origin and $P(x,y,z)$ be any point, then $OP=\sqrt{({{x}^{2}}+{{y}^{2}}+{{z}^{2}})}$.   (3) Distance of a point from co-ordinate axes : Let $P(x,y,z)$ be any point in the space. Let $PA,PB$ and $PC$ be the perpendiculars drawn from P to the axes $OX,\,\,\,OY$ and $OZ$ respectively.       Then,       $PA=\sqrt{({{y}^{2}}+{{z}^{2}})}$ $PB=\sqrt{({{z}^{2}}+{{x}^{2}})}$  $PC=\sqrt{({{x}^{2}}+{{y}^{2}})}$

#### Properties of Some Geometrical Figures

(1) In a triangle ABC, if AD is the median drawn to BC, then $A{{B}^{2}}+A{{C}^{2}}=2(A{{D}^{2}}+B{{D}^{2}})$     (2) A triangle is isosceles if any two of its medians are equal or two sides are equal.     (3) In a right angled triangle, the mid-point of the hypotenuse is equidistant from the vertices.     (4) Equilateral triangle : All sides are equal.     (5) Rhombus : All sides are equal and no angle is right angle, but diagonals are at right angles and unequal.     (6) Square : All sides are equal and each angle is right angle. The diagonals bisect each other.     (7) Parallelogram : Opposite sides are parallel and equal and diagonals bisect each other.     (8) Rectangle : Opposite sides are equal and each angle is right angle. Diagonals are equal.     (9) The figure obtained by joining the middle points of a quadrilateral in order is a parallelogram.

#### Co-ordinates of a Point in Space

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

#### Distance Formula

The distance between two points $P({{x}_{1}},{{y}_{1}})$ and $Q({{x}_{2}},{{y}_{2}})$ is given by $PQ=\sqrt{{{(PR)}^{2}}+{{(QR)}^{2}}}=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}$   Distance between two points in polar co-ordinates :    Let O be the pole and OX  be the initial line. Let P and Q be two given points whose polar co-ordinates are $({{r}_{1}},{{\theta }_{1}})$ and $({{r}_{2}},{{\theta }_{2}})$ respectively.              Then,  ${{(PQ)}^{2}}=r_{1}^{2}+r_{2}^{2}-2{{r}_{1}}{{r}_{2}}\cos ({{\theta }_{1}}-{{\theta }_{2}})$     $\therefore$       $PQ=\sqrt{r_{1}^{2}+r_{2}^{2}-2{{r}_{1}}{{r}_{2}}\cos ({{\theta }_{1}}-{{\theta }_{2}})}$,     where ${{\theta }_{1}}$ and ${{\theta }_{2}}$ in radians.

#### Polar Co-ordinates

Let $OX$ be any fixed line which is usually called the initial line and O be a fixed point on it. If distance of any point P from the O is $'r'$ and $\angle XOP=\theta$, then $(r,\,\,\theta )$are called the polar co-ordinates of a point P.           If $(x,\,\,y)$ are the cartesian co-ordinates of a point P, then    $x=r\,\cos \theta ;\,\,y=r\sin \theta$   and $r=\sqrt{{{x}^{2}}+{{y}^{2}}};\,\,\,\theta ={{\tan }^{-1}}\left( \frac{y}{x} \right)$.

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