KEAM – Mathematics
Topic: Planes and Lines in 3‑D
Goal: Understand the algebraic description of straight lines and planes in 3‑D, how to find distances, angles, intersections and to solve typical exam problems.
1. Simple Explanation
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Line in 3‑D – A set of points that can be described by a point (P_0(x_0,y_0,z_0)) and a direction vector (\vec d = (l,m,n)).
[ \boxed{\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}} ] (Symmetric form) or parametric:
[ x = x_0 + l t,; y= y_0 + m t,; z = z_0 + n t ] -
Plane in 3‑D – A set of points that can be described by a point (P_0(x_0,y_0,z_0)) and a normal vector (\vec n = (A,B,C)).
[ \boxed{A(x-x_0)+B(y-y_0)+C(z-z_0)=0} ] (Cartesian form)
[ \boxed{Ax+By+Cz+D=0} ] -
Direction Ratios (DR) – The components of a direction vector.
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Normal Vector – Perpendicular to the plane (or direction vector of a line).
2. Key Points
| Concept | Formula | Remarks |
|---|---|---|
| Line Direction Ratios | (\vec d = (l,m,n)) | (l:m:n) can be any proportional set. |
| Line Parametric/Symmetric | ( \frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}) | If one ratio is 0, replace with (x=x_0) etc. |
| Plane Cartesian | (Ax+By+Cz+D=0) | (A,B,C) are normal components. |
| Plane Normal | (\vec n=(A,B,C)) | ( |
| Point–Plane Distance | (d=\frac{ | Ax_1+By_1+Cz_1+D |
| Distance Between Parallel Planes | (d=\frac{ | D_2-D_1 |
| Distance Between a Point & a Line | (d=\frac{ | \vec{P_0P}\times \vec d |
| Angle Between Line & Plane | (\sin\theta = \frac{ | \vec d\cdot\vec n |
| Angle Between Two Planes | (\cos\phi=\frac{ | \vec n_1\cdot\vec n_2 |
| Intersection of Line & Plane | Substitute line parametric into plane equation → solve for (t). | |
| Intersection of Two Planes | Solve linear system of two plane equations → line of intersection. | |
| Skew Lines | No common point; not parallel. | Shortest distance formula below. |
| Shortest Distance Between Skew Lines | [ | |
| d = \frac{ | (\vec P_1-\vec P_2)\cdot(\vec d_1\times\vec d_2) | } |
| { | \vec d_1\times\vec d_2 | } |
| ] | (\vec P_1,\vec P_2) are points on each line. | |
| Parallel Lines | Direction ratios proportional: (\vec d_1 \parallel \vec d_2). | |
| Perpendicular Lines | (\vec d_1\cdot\vec d_2 = 0). | |
| Line Parallel to Plane | (\vec d\cdot\vec n = 0). | |
| Line Perpendicular to Plane | (\vec d \parallel \vec n). |
3. Worked Example
Problem
Find the distance between the skew lines
[ \mathcal{L}_1:;\frac{x-1}{2}=\frac{y-0}{-1}=\frac{z-3}{1} \qquad\text{and}\qquad \mathcal{L}_2:;x=0,; y=2,; z=1+2t ]
Step‑by‑Step Solution
- Convert to parametric form
[ \mathcal{L}_1:; \begin{cases} x = 1 + 2s\ y = 0 - s\ z = 3 + s \end{cases} \quad (s\in\mathbb R) ] [ \mathcal{L}_2:; \begin{cases} x = 0\ y = 2