\displaystyle \textbf{1.} \ \text{If }P(x_{1},y_{1},z_{1})\text{ and }Q(x_{2},y_{2},z_{2})\text{ are two points in space, then}
\displaystyle PQ=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}

\displaystyle \textbf{2.} \ \text{The distance of a point }P(x,y,z)\text{ from the origin }O\text{ is given by }OP=\sqrt{x^{2}+y^{2}+z^{2}}

\displaystyle \textbf{3.} \ \text{If }P(x_{1},y_{1},z_{1})\text{ and }Q(x_{2},y_{2},z_{2})\text{ are two points, then the coordinates of a point }PQ
\displaystyle \text{dividing internally in the ratio }m:n\text{ are }\left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n},\frac{mz_{2}+nz_{1}}{m+n}\right)
\displaystyle \text{If }R\text{ divides }PQ\text{ externally in the ratio }m:n,\text{ then its coordinates are}
\displaystyle \left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n},\frac{mz_{2}-nz_{1}}{m-n}\right)
\displaystyle \text{The coordinates of the mid-point of }PQ\text{ are }\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2}\right)

\displaystyle \textbf{4.} \ \text{The line segment joining }P(x_{1},y_{1},z_{1})\text{ and }Q(x_{2},y_{2},z_{2})\text{ is divided by}
\displaystyle (i)\ \text{YZ-plane in the ratio }-x_{1}:x_{2}\qquad (ii)\ \text{ZX-plane in the ratio }-y_{1}:y_{2}
\displaystyle (iii)\ \text{XY-plane in the ratio }-z_{1}:z_{2}

\displaystyle \textbf{5.} \ \text{The coordinates of the centroid of the triangle formed by the points }(x_{1},y_{1},z_{1}) \\ \text{and }(x_{2},y_{2},z_{2})\text{ 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)

\displaystyle \textbf{6.} \ \text{The coordinates of the centroid of the tetrahedron formed by the points } \\ (x_{1},y_{1},z_{1}),\ (x_{2},y_{2},z_{2}),  (x_{3},y_{3},z_{3})\text{ and }(x_{4},y_{4},z_{4})\text{ 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)

\displaystyle \textbf{7.} \ \text{The distances of point }P(x,y,z)\text{ from }x,\ y\text{ and }z\text{ axes are }\sqrt{y^{2}+z^{2}},\sqrt{z^{2}+x^{2}} \\ \text{ and }\sqrt{x^{2}+y^{2}}\text{ respectively.}

\displaystyle \textbf{8.} \ \text{If a directed line segment }OP\text{ makes angles }\alpha,\beta,\gamma\text{ with }OX,\ OY \text{ and }OZ \\ \text{respectively, } \text{then }\cos\alpha,\cos\beta,\cos\gamma\text{ are known as the direction cosines of } OP\text{ and are} \\ \text{generally denoted by }l,m,n.
\displaystyle \text{Thus, we have }l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma
\displaystyle \text{Direction cosines of }PO\text{ are }-l,-m,-n
\displaystyle \text{If }OP=r\text{ and the coordinates of }P\text{ are }(x,y,z),\text{ then }x=lr,\ y=mr, \\ \ z=nr

\displaystyle \textbf{9.} \ \text{If }l,m,n\text{ are direction cosines of a vector }\overrightarrow{r},\text{ then}
\displaystyle (i)\ \overrightarrow{r}=|\overrightarrow{r}|(l\widehat{i}+m\widehat{j}+n\widehat{k})\text{ and }\widehat{r}=l\widehat{i}+m\widehat{j}+n\widehat{k}
\displaystyle (ii)\ l^{2}+m^{2}+n^{2}=1
\displaystyle (iii)\ \text{Projections of }\overrightarrow{r}\text{ on the coordinate axes are }l|\overrightarrow{r}|,\ m|\overrightarrow{r}|,\ n|\overrightarrow{r}|
\displaystyle (iv)\ |\overrightarrow{r}|=\sqrt{\text{Sum of the squares of projections of }\overrightarrow{r}\text{ on the coordinate axes}}

\displaystyle \textbf{10.} \ \text{If }P(x_{1},y_{1},z_{1})\text{ and }Q(x_{2},y_{2},z_{2})\text{ are two points such that the direction cosines of } \\ \overrightarrow{PQ}\text{ are }l,m,n,
\displaystyle \text{then }x_{2}-x_{1}=l|\overrightarrow{PQ}|,\ y_{2}-y_{1}=m|\overrightarrow{PQ}|,\ z_{2}-z_{1}=n|\overrightarrow{PQ}|
\displaystyle \text{These are projections of }\overrightarrow{PQ}\text{ on }X,Y\text{ and }Z\text{-axes respectively.}

\displaystyle \textbf{11.} \ \text{If }l,m,n\text{ are direction cosines of a vector }\overrightarrow{r}\text{ and }a,b,c\text{ are three numbers such that } \\ \frac{l}{a}=\frac{m}{b}=\frac{n}{c},
\displaystyle \text{then, we say that the direction ratios of }\overrightarrow{r}\text{ are proportional to }a,b,c.
\displaystyle \text{Also, }l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}},\ m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}},\ n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}

\displaystyle \textbf{12.} \ \text{If }\theta\text{ is the angle between two lines having direction cosines }l_{1},m_{1},n_{1}\text{ and }l_{2},m_{2},n_{2},\text{ then}
\displaystyle \cos\theta=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}
\displaystyle (i)\ \text{Lines are parallel, iff }\frac{l_{1}}{l_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}
\displaystyle (ii)\ \text{Lines are perpendicular, iff }l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}=0

\displaystyle \textbf{13.} \ \text{If }\theta\text{ is the angle between two lines whose direction ratios are proportional to }a_{1},b_{1},c_{1} \\ \text{ and }a_{2},b_{2},c_{2}\text{ respectively, then}
\displaystyle \cos\theta=\frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}
\displaystyle \text{Lines are parallel, iff }\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}
\displaystyle \text{Lines are perpendicular, iff }a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}=0

\displaystyle \textbf{14.} \ \text{The projection of the line segment joining points }P(x_{1},y_{1},z_{1})\text{ and } Q(x_{2},y_{2},z_{2}) \\ \text{ to the line having direction cosines }l,m,n\text{ is}
\displaystyle (x_{2}-x_{1})l+(y_{2}-y_{1})m+(z_{2}-z_{1})n

\displaystyle \textbf{15.} \ \text{The direction ratios of the line passing through points }P(x_{1},y_{1},z_{1})\text{ and} \\ Q(x_{2},y_{2},z_{2}) \text{ are proportional to }x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1}
\displaystyle \text{Direction cosines of }\overrightarrow{PQ}\text{ are }\frac{x_{2}-x_{1}}{PQ},\frac{y_{2}-y_{1}}{PQ},\frac{z_{2}-z_{1}}{PQ}

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