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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 \[={{z}_{2}}{{z}_{1}}\].  

(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., \[l=\frac{{{x}_{2}}-{{x}_{1}}}{PQ},\,m=\frac{{{y}_{2}}-{{y}_{1}}}{PQ},\,n=\frac{{{z}_{2}}-{{z}_{1}}}{PQ}\].

(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 \[V=\frac{1}{6}\left| \,\begin{matrix} {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1  \\ {{x}_{4}} & {{y}_{4}} & {{z}_{4}} & 1  \\ \end{matrix}\, \right|\].

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 \[y=\frac{{{m}_{1}}{{y}_{2}}-{{m}_{2}}{{y}_{1}}}{{{m}_{1}}-{{m}_{2}}}\]

(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 \[PQ\].

(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,       \[PA=\sqrt{({{y}^{2}}+{{z}^{2}})}\] \[PB=\sqrt{({{z}^{2}}+{{x}^{2}})}\]  \[PC=\sqrt{({{x}^{2}}+{{y}^{2}})}\]  

(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.

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

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.

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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