\displaystyle 1.\ \text{Two non-parallel planes always intersect in a straight line. Thus, if }a_{1}x+b_{1}y+c_{1}z+d_{1}=0
\displaystyle \text{ and }a_{2}x+b_{2}y+c_{2}z+d_{2}=0\text{ are equations of two non-parallel planes, then these two}
\displaystyle \text{equations taken together represent a line.}
\displaystyle \text{i.e. }a_{1}x+b_{1}y+c_{1}z+d_{1}=0=a_{2}x+b_{2}y+c_{2}z+d_{2}\text{ is the equation of a line.}
\displaystyle \text{This is known as an un-symmetrical form of a line.}

\displaystyle 2.\ \text{The equations of a line passing through a point }(x_{1},y_{1},z_{1})\text{ and having direction cosines (or}
\displaystyle \text{direction ratios) }l,m,n\text{ are given by }\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}.
\displaystyle \text{The coordinates of an arbitrary point on this line are }(x_{1}+lr,\ y_{1}+mr,\ z_{1}+nr),\text{ where }r\text{ is a}\text{parameter. This is known as symmetrical form of a line.}

\displaystyle 3.\ \text{The vector equation of a line passing through a point having position vector }\overrightarrow{a}\text{ and parallel}
\displaystyle \text{to vector }\overrightarrow{b}\text{ is }\overrightarrow{r}=\overrightarrow{a}+\lambda\overrightarrow{b},\text{ where }\lambda\text{ is a parameter.}

\displaystyle 4.\ \text{The equations of a line passing through points }(x_{1},y_{1},z_{1})\text{ and }(x_{2},y_{2},z_{2})\text{ are given by}
\displaystyle \frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}.}

\displaystyle 5.\ \text{The vector equation of a line passing through points having position vectors }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ is}
\displaystyle \overrightarrow{r}=\overrightarrow{a}+\lambda(\overrightarrow{b}-\overrightarrow{a}),\text{ where }\lambda\text{ is a parameter.}

\displaystyle 6.\ \text{If }l,m,n\text{ are the direction cosines of the line of intersection of planes }a_{1}x+b_{1}y+c_{1}z+d_{1}=0\text{ and }a_{2}x+b_{2}y+c_{2}z+d_{2}=0,\text{ then}
\displaystyle a_{1}l+b_{1}m+c_{1}n=0\text{ and }a_{2}l+b_{2}m+c_{2}n=0
\displaystyle \therefore \frac{l}{b_{1}c_{2}-b_{2}c_{1}}=\frac{m}{c_{1}a_{2}-c_{2}a_{1}}=\frac{n}{a_{1}b_{2}-a_{2}b_{1}}
\displaystyle \text{So, }l,m,n\text{ are proportional to }b_{1}c_{2}-b_{2}c_{1},\ c_{1}a_{2}-c_{2}a_{1},\ a_{1}b_{2}-a_{2}b_{1}.

\displaystyle 7.\ (i)\ \text{The length of the perpendicular from a point }P(\overrightarrow{\alpha})\text{ on the line }\overrightarrow{r}=\overrightarrow{a}+\lambda\overrightarrow{b}\text{ is given by}
\displaystyle \sqrt{|\overrightarrow{\alpha}-\overrightarrow{a}|^{2}-\left\{\frac{(\overrightarrow{\alpha}-\overrightarrow{a})\cdot\overrightarrow{b}}{|\overrightarrow{b}|}\right\}^{2}}
\displaystyle (ii)\ \text{The length of the perpendicular from a point }P(x_{1},y_{1},z_{1})\text{ on the line }\frac{x-a}{l}=\frac{y-b}{m}=\frac{z-c}{n}\text{ is given by}
\displaystyle \sqrt{(a-x_{1})^{2}+(b-y_{1})^{2}+(c-z_{1})^{2}-\{(a-x_{1})l+(b-y_{1})m+(c-z_{1})n\}^{2}}
\displaystyle \text{where }l,m,n\text{ are direction cosines of the line.}

\displaystyle 8.\ \text{Two straight lines in space are said to be skew lines if they are neither parallel} \\ \text{nor intersecting.}

\displaystyle 9.\ \text{If }l_{1}\text{ and }l_{2}\text{ are two skew lines, then a line perpendicular to each of } l_{1}\text{ and }l_{2} \\ \text{ is known as the line of shortest distance.}
\displaystyle \text{If the line of shortest distance intersects lines }l_{1}\text{ and }l_{2} \text{ at }P\text{ and }Q\text{ respectively,} \\ \text{then the distance } PQ\text{ between points }P\text{ and } Q\text{ is known as the shortest distance} \\ \text{between }l_{1}\text{ and }l_{2}.

\displaystyle 10.\ \text{The shortest distance between lines }\overrightarrow{r}=\overrightarrow{a_{1}}+\lambda\overrightarrow{b_{1}}\text{ and }\overrightarrow{r}=\overrightarrow{a_{2}}+\mu\overrightarrow{b_{2}}\text{ is given by}
\displaystyle d=\frac{|(\overrightarrow{b_{1}}\times\overrightarrow{b_{2}})\cdot(\overrightarrow{a_{2}}-\overrightarrow{a_{1}})|}{|\overrightarrow{b_{1}}\times\overrightarrow{b_{2}}|}

\displaystyle 11.\ \text{The shortest distance between the lines }\frac{x-x_{1}}{l_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}}\text{ and }\frac{x-x_{2}}{l_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}}\text{ is given by}
\displaystyle d=\frac{\begin{vmatrix}x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\l_{1}&m_{1}&n_{1}\\l_{2}&m_{2}&n_{2}\end{vmatrix}}{\sqrt{(m_{1}n_{2}-m_{2}n_{1})^{2}+(n_{1}l_{2}-n_{2}l_{1})^{2}+(l_{1}m_{2}-l_{2}m_{1})^{2}}}

\displaystyle 12.\ \text{The shortest distance between parallel lines }\overrightarrow{r}=\overrightarrow{a_{1}}+\lambda\overrightarrow{b}\text{ and }\overrightarrow{r}=\overrightarrow{a_{2}}+\mu\overrightarrow{b}\text{ is given by} d=\frac{|(\overrightarrow{a_{1}}-\overrightarrow{a_{2}})\times\overrightarrow{b}|}{|\overrightarrow{b}|}

\displaystyle 13.\ \text{Lines }\overrightarrow{r}=\overrightarrow{a_{1}}+\lambda\overrightarrow{b_{1}}\text{ and }\overrightarrow{r}=\overrightarrow{a_{2}}+\mu\overrightarrow{b_{2}}\text{ are intersecting lines, iff }\\ (\overrightarrow{b_{1}}\times\overrightarrow{b_{2}})\cdot(\overrightarrow{a_{2}}-\overrightarrow{a_{1}})=0.

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