## Three - dimensional Geometry

# Equation of a Line in Space

- The equation of the straight line passing through the point A(\vec{a}) and parallel to the vector B(\vec{b}) is \vec{r}=\vec{a}+\lambda \vec{b},, where λ is scalar.
- The equation of the straight line passing through the point A(x
_{1}, y_{1}, z_{1}) and direction ratios proportional to the line a, b, c are \frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}=\lambda - The equation of the straight line passing through the points A(\vec{a}) and B(\vec{b}) is \vec{r}=\vec{a}+\lambda (\vec{b}-\vec{a}) (or) \vec{r}=(1-\lambda)\vec{a}+\lambda \vec{b}
- The equation of the straight line joining two points A(x
_{1}, y_{1}, z_{1}) B(x_{2}, y_{2}, z_{2}) is \frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}=\lambda - If the straight lines whose directions cosines l, m, n are given by al + bm + cn = 0, fmn + gnl + hlm = 0 are

(i) Perpendicular if \frac{f}{a} + \frac{g}{b} + \frac{h}{c} = 0

(ii) Parallel if \sqrt{af}\pm \sqrt{bg} \pm \sqrt{ch}= 0 - If the straight lines whose direction cosines are given by the equations al + bm + cn = 0, ul
^{2}+ vm^{2}+ wn^{2}= 0 are

(i) Perpendicular if (v + w) a^{2}+ (w + u) b^{2}+ (u + v) c^{2}= 0

(ii) Parallel if \frac{a^{2}}{u}+\frac{b^{2}}{v}+\frac{c^{2}}{w}=0 - The projections of the line joining two points A(x
_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) on the line whose direction cosines are l, m, n is |l (x_{2 }− x_{1}) + m (y_{2 }− y_{1}) + n (z_{2 }− z_{1})| - If l, m, n are the direction cosines of a line which is perpendicular to two mutually perpendicular lines whose direction cosines are (l
_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) then l = m_{1}n_{2}− m_{2 }n_{1}, m = n_{1}l_{2}− n_{2 }l_{1}, n = l_{1}m_{2}− l_{2 }m_{1} - Two lines \vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1} and \vec{r}=\vec{a}_{2}+\mu \vec{b}_{2} are

(i) Parallel \Rightarrow\vec{b}_{1}\ ||\ \vec{b}_{2}

(ii) Perpendicular \Rightarrow\vec{b}_{1}\ \cdot\ \vec{b}_{2} = 0 - Two lines \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}} and \frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{b_{2}}=\frac{z-z_{2}}{c_{2}} are

(i) Parallel \Rightarrow\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

(ii) Perpendicular ⇒ a_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}= 0

### View the Topic in this video From 37:20 To 54:42

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1. Vector equation of a line that passes through the given point whose position vector is \vec{a} and parallel to a given vector \vec{b} is \vec{r}=\vec{a}+\lambda\vec{b}

2. Equation of a line through a point (x_{1}, y_{1}, z_{1}) and having direction cosines l, m, n is \frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}

3. The vector equation of a line which passes through two points whose position vectors are \vec{a} and \vec{b} is \vec{r}=\vec{a}+\lambda (\vec{b}-\vec{a}).

4. Cartesian equation of a line that passes through two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is \frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}