Line and Plane

1. Equations of a Line

• Passing through a point $A(\overline{a})$ and parallel to vector $\overline{b}$

  Vector form: $\overline{r} = \overline{a} + \lambda\overline{b}$

  Non-parametric form: $(\overline{r} - \overline{a}) \times \overline{b} = \overline{0}$

  Cartesian form: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda$ (Symmetric form)

  Line passing through a point and parallel to a vector
• Passing through two points $A(\overline{a})$ and $B(\overline{b})$

  Vector form: $\overline{r} = \overline{a} + \lambda(\overline{b} - \overline{a})$

  Non-parametric form: $(\overline{r} - \overline{a}) \times (\overline{b} - \overline{a}) = \overline{0}$

  Cartesian form: $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$

2. Distance Involving Lines

• Distance of point $P(\overline{\alpha})$ from line $\overline{r} = \overline{a} + \lambda\overline{b}$

  $$d = \sqrt{|\overline{\alpha} - \overline{a}|^2 - \left[\frac{(\overline{\alpha} - \overline{a}) \cdot \overline{b}}{|\overline{b}|}\right]^2}$$

  Perpendicular distance from a point to a line
• Shortest distance between Skew Lines

  For lines $\overline{r} = \overline{a_1} + \lambda_1\overline{b_1}$ and $\overline{r} = \overline{a_2} + \lambda_2\overline{b_2}$:

  $$d = \left| \frac{(\overline{a_2} - \overline{a_1}) \cdot (\overline{b_1} \times \overline{b_2})}{|\overline{b_1} \times \overline{b_2}|} \right|$$

• Intersection Condition

  Two lines intersect each other if and only if the shortest distance between them is zero:

  $(\overline{a_2} - \overline{a_1}) \cdot (\overline{b_1} \times \overline{b_2}) = 0$

• Distance between Parallel Lines

  For lines $\overline{r} = \overline{a_1} + \lambda\overline{b}$ and $\overline{r} = \overline{a_2} + \lambda\overline{b}$:

  $$d = |(\overline{a_2} - \overline{a_1}) \times \hat{b}|$$

3. Equations of a Plane

• Passing through point $A(\overline{a})$ and perpendicular to normal $\overline{n}$

  Vector form: $\overline{r} \cdot \overline{n} = \overline{a} \cdot \overline{n}$ (or $\overline{r} \cdot \overline{n} = d$)

  Cartesian form: $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$

  Equation of a plane passing through a point and perpendicular to a vector
• Normal form of Plane

  Vector form: $\overline{r} \cdot \hat{n} = p$ (where $p$ is the distance from the origin)

  Cartesian form: $lx + my + nz = p$

  Foot of perpendicular from origin: $N(pl, pm, pn)$

• Passing through $A(\overline{a})$ and parallel to non-collinear vectors $\overline{b}$ and $\overline{c}$

  Parametric form: $\overline{r} = \overline{a} + \lambda\overline{b} + \mu\overline{c}$

  Non-parametric form: $\overline{r} \cdot (\overline{b} \times \overline{c}) = \overline{a} \cdot (\overline{b} \times \overline{c})$

• Passing through three non-collinear points $A(\overline{a}), B(\overline{b}), C(\overline{c})$

  Vector form: $(\overline{r} - \overline{a}) \cdot (\overline{b} - \overline{a}) \times (\overline{c} - \overline{a}) = 0$

  Cartesian form: $$\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$$

• Passing through intersection of two planes

  Vector form: $\overline{r} \cdot (\overline{n_1} + \lambda\overline{n_2}) - (d_1 + \lambda d_2) = 0$

  Cartesian form: $(a_1x+b_1y+c_1z+d_1) + \lambda(a_2x+b_2y+c_2z+d_2) = 0$

4. Angles and Distances Involving Planes

• Angle between two planes

  For planes $\overline{r} \cdot \overline{n_1} = d_1$ and $\overline{r} \cdot \overline{n_2} = d_2$:

  $$\cos \theta = \left| \frac{\overline{n_1} \cdot \overline{n_2}}{|\overline{n_1}| |\overline{n_2}|} \right|$$

• Angle between a line and a plane

  For line $\overline{r} = \overline{a} + \lambda\overline{b}$ and plane $\overline{r} \cdot \overline{n} = d$:

  $$\sin \theta = \left| \frac{\overline{b} \cdot \overline{n}}{|\overline{b}| |\overline{n}|} \right|$$

• Distance of a point from a plane

  The distance of point $A(\overline{a})$ from plane $\overline{r} \cdot \hat{n} = p$:

  $$d = |p - \overline{a} \cdot \hat{n}|$$

5. Coplanarity of Two Lines

• Condition for Coplanarity

  Lines $\overline{r} = \overline{a_1} + \lambda_1\overline{b_1}$ and $\overline{r} = \overline{a_2} + \lambda_2\overline{b_2}$ are coplanar if and only if:

  $$(\overline{a_2} - \overline{a_1}) \cdot (\overline{b_1} \times \overline{b_2}) = 0$$

• Equation of the Plane

  The equation of the plane determined by the coplanar lines is:

  $$(\overline{r} - \overline{a_1}) \cdot (\overline{b_1} \times \overline{b_2}) = 0$$

• Cartesian Form Condition

  $$\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0$$