 # 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 (x− x1) + m (y− y1) + n (z− 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 − mn1 , m = n1l2 − nl1 , n = l1m2 − lm1
• 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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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}}