Vectors & 3D Geometry
Study Notes for ITI Admission Test – Mathematics
1. Simple Explanation
| Concept | What it means | Why it matters |
|---|---|---|
| Vector | A directed line segment with magnitude (length) and direction. Represented as <x, y, z> in 3‑D. | Gives a concise way to describe motion, forces, positions, etc. |
| Dot product | Measure of how much two vectors point in the same direction. | Helps find angles, projections, and work done by a force. |
| Cross product | A vector perpendicular to two given vectors; magnitude = area of parallelogram they span. | Used to find normals, torque, area, and orientation. |
| Line (3‑D) | Set of points that satisfy a direction vector and a point on the line. | Describes paths, beams, etc. |
| Plane (3‑D) | Set of points that satisfy a normal vector and a point on the plane. | Models surfaces, walls, etc. |
| Distance | Shortest distance between a point and a line/plane. | Useful for engineering, navigation, and geometry problems. |
2. Key Points
| Topic | Key Formulas & Properties |
|---|---|
| Vector Arithmetic | Addition: <a, b, c> + <d, e, f> = <a+d, b+e, c+f> Scalar Multiplication: k<a, b, c> = <ka, kb, kc> |
| Magnitude (Length) | |
| Unit Vector | û = v / |
| Dot Product | v·w = x₁x₂ + y₁y₂ + z₁z₂ v·w = |
| Cross Product | v × w = <y₁z₂ − z₁y₂, z₁x₂ − x₁z₂, x₁y₂ − y₁x₂> |
| Vector Form of a Line | r = r₀ + λd (r₀ = point on line, d = direction) |
| Symmetric Form | (x−x₀)/dₓ = (y−y₀)/dᵧ = (z−z₀)/d_z = λ |
| Plane Equation (Normal Form) | n·(r−r₀) = 0 ⇔ A(x−x₀)+B(y−y₀)+C(z−z₀)=0 |
| Distance from Point P(x₀,y₀,z₀) to Plane Ax+By+Cz+D=0 | d = |
| Distance from Point to Line | Let line: r = a + λd. d = |
| Intersection of Two Planes | Solve the two plane equations simultaneously; if normals are not parallel, intersection is a line. |
| Area of Parallelogram spanned by v,w |
3. Worked Example
Problem
Find:
- The equation of the line through points
(P(1,,2,,3)) and (Q(4,,0,,6)). - The equation of the plane that contains point (R(2,,1,,0)) and is perpendicular to the vector (\mathbf{n} = \langle 3,,1,,-2\rangle).
- The shortest distance from point (S(5,,2,,1)) to that plane.
- The area of the parallelogram formed by vectors (\mathbf{a}=\langle 1,,0,,2\rangle) and (\mathbf{b}=\langle 0,,3,,1\rangle).
1. Line through (P) and (Q)
Direction vector ( \mathbf{d} = Q-P = \langle 4-1,,0-2,,6-3\rangle = \langle 3,,-2,,3\rangle.)
Vector form
[
\boxed{;\mathbf{r} = \langle 1,,2,,3\rangle + \lambda,\langle 3,,-2,,3\rangle;}
]
Parametric equations
[
x = 1 + 3\lambda,\quad y = 2 - 2\lambda,\quad z = 3 + 3\lambda.
]
Symmetric form
[
\frac{x-1}{3}=\frac{y-2}{-2}=\frac{z-3}{3}.
]