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(x1, y1, z1) 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(x1, y1, z1) B(x2, y2, z2) 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, ul2 + vm2 + wn2 = 0 are 
    (i) Perpendicular if (v + w) a2 + (w + u) b2 + (u + v) c2 = 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(x1, y1, z1) and B(x2, y2, z2) on the line whose direction cosines are l, m, n is |l (x2 − x1) + m (y2 − y1) + n (z2 − z1)| 
  • If l, m, n are the direction cosines of a line which is perpendicular to two mutually perpendicular lines whose direction cosines are (l1, m1, n1) and (l2, m2, n2) then l = m1n2 − m2 n1 , m = n1l2 − n2 l1 , n = l1m2 − l2 m1
  • 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 ⇒ a1a2 + b1b2 + c1c2 = 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 (x1, y1, z1) 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 (x1, y1, z1) and (x2, y2, z2) 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}}