Definitions [1]
An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.
Formulae [5]
\[SD=\left|\frac{\left(a_{2}-a_{1}\right)\times b}{\left|b\right|}\right|\]
Vector Form:
\[\mathbf{d}=\left|\frac{(\overline{\mathbf{b}}_{1}\times\overline{\mathbf{b}}_{2}).(\overline{\mathbf{a}}_{2}-\overline{\mathbf{a}}_{1})}{\left|\overline{\mathbf{b}}_{1}\times\overline{\mathbf{b}}_{2}\right|}\right|\]
Cartesian Form:
\[\mathbf{d}=\left|\frac{ \begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\ \mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \end{vmatrix}}{\sqrt{\left(\mathbf{a}_1\mathbf{b}_2-\mathbf{a}_2\mathbf{b}_1\right)^2+\left(\mathbf{a}_1\mathbf{c}_2-\mathbf{a}_2\mathbf{c}_1\right)^2+\left(\mathbf{b}_1\mathbf{c}_2-\mathbf{b}_2\mathbf{c}_1\right)^2}}\right|\]
From general form:
- Slope (m) = −a / b
- Y-intercept = −c / b
Vector Form:
\[\mathbf{d}=\frac{\left|\left(\overline{\mathbf{a}}.\overline{\mathbf{n}}\right)-\mathbf{p}\right|}{\left|\overline{\mathbf{n}}\right|}\]
Cartesian Form:
\[\mathbf{d}=\left|\frac{\mathbf{a}x_{1}+\mathbf{b}y_{1}+\mathbf{c}z_{1}+\mathbf{d}}{\sqrt{\mathbf{a}^{2}+\mathbf{b}^{2}+\mathbf{c}^{2}}}\right|\]
Vector:
Angle between two lines: \[\cos\theta=\left|\frac{\mathbf{b}_{1}\cdot\mathbf{b}_{2}}{|\mathbf{b}_{1}||\mathbf{b}_{2}|}\right|\]
If two lines are perpendicular: b₁ · b₂ = 0
If two lines are parallel: b₁ = λb₂
Cartesian:
\[\cos\theta=\frac{|a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]
If two lines are perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
If two lines are parallel: \[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]
Key Points
| Form | Formula |
|---|---|
| X-axis | y = 0 |
| Y-axis | x = 0 |
| Parallel to the X-axis | y = b or y = -b |
| Parallel to the Y-axis | x = a or x = -a |
| Slope-point form | y − y₁ = m(x − x₁) |
| Two-point form | \[\frac{y-y_{1}}{y_{1}-y_{2}}=\frac{x-x_{1}}{x_{1}-x_{2}}\] |
| Slope-intercept form | y = mx + c |
| Intercept form | \[\frac{x}{\mathrm{a}}+\frac{y}{\mathrm{b}}=1\] |
| Normal form | x cosα + y sinα = p |
| Parametric form | \[\frac{x-x_{1}}{\cos\theta}=\frac{y-y_{1}}{\sin\theta}=r\] |
Position of a Point:
For line: ax₁ + by₁ + c
- If ax₁ + by₁ + c = 0 → Point lies on the line
- If ax₁ + by₁ + c < 0 → Point lies on one side (origin side)
- If ax₁ + by₁ + c > 0 → Point lies on other side
Vector Form:
Condition for coplanarity of two lines:
Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if
(a₁ − a₂) · (b₁ × b₂) = 0
Equation of the plane containing both lines:
\[\left(\overline{\mathbf{r}}-\overline{\mathbf{a}_1}\right).\left(\overline{\mathbf{b}_1}\times\overline{\mathbf{b}_2}\right)=\mathbf{0}\] or \[\left(\overline{\mathbf{r}}-\overline{\mathbf{a}_2}\right).\left(\overline{\mathbf{b}_1}\times\overline{\mathbf{b}_2}\right)=\mathbf{0}\]
Cartesian Form:
\[\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\ \mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \end{vmatrix}=0\]
| Case | Vector Form | Cartesian Form (Symmetric Form) |
|---|---|---|
| 1. Through a point + parallel to vector | r = a + λb | x = x₁ + lλ y = y₁ + mλ z = z₁ + nλ |
| 2. Through two points | r = a + λ(b − a) | x − x₁ / (x₂ − x₁) = y − y₁ / (y₂ − y₁) = z − z₁ / (z₂ − z₁) |
Concepts [16]
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Equation of a Plane Passing Through Three Non Collinear Points
- Equations of Line in Different Forms
- Plane Passing Through the Intersection of Two Given Planes
- Coplanarity of Two Lines
- Angle Between Two Planes
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Vector and Cartesian Equation of a Plane
- Vector and Cartesian Equations of a Line
