Vectors & 3D Geometry
State Polytechnic Entrance – Mathematics
These notes give a concise, self‑contained overview of vectors, dot & cross products, and the geometry of lines and planes in three dimensions.
1. Simple Explanation
| Concept | What it Is | How to Think About It |
|---|---|---|
| Vector | A directed line segment, described by its magnitude (length) and direction. | Think of a force acting from one point to another. |
| Coordinate Representation | In 3‑D a vector v = ⟨v₁, v₂, v₃⟩. | The three components are the changes in x, y, and z. |
| Vector Addition | u + v = ⟨u₁+v₁, u₂+v₂, u₃+v₃⟩. | “Tail‑to‑head” or component‑wise. |
| Scalar Multiplication | k v = ⟨k v₁, k v₂, k v₃⟩. | Stretch or shrink the vector. |
| Magnitude | v | |
| Dot Product | u·v = u₁v₁+u₂v₂+u₃v₃. | Measures how “parallel” two vectors are. |
| Cross Product | u×v = ⟨u₂v₃−u₃v₂, u₃v₁−u₁v₃, u₁v₂−u₂v₁⟩. | Gives a vector perpendicular to both u and v. |
| Line (3‑D) | Parametric: r = a + tb. Symmetric: (x−aₓ)/bₓ = (y−aᵧ)/bᵧ = (z−a_z)/b_z. |
a is a point on the line, b is a direction vector. |
| Plane (3‑D) | Normal form: n·(r−a) = 0. General form: Ax+By+Cz+D = 0. |
a is a point on the plane, n is a normal vector. |
| Distance | From point P to plane: | P−A |
2. Key Points
| Item | Formula / Property | Notes |
|---|---|---|
| Vector basics | v = ⟨v₁,v₂,v₃⟩, | v |
| Unit vector | u = v / | v |
| Dot product | u·v = | u |
| Orthogonality | u·v = 0 | Vectors are perpendicular. |
| Cross product | u×v | |
| Triple scalar product | (a×b)·c | Volume of parallelepiped. |
| Line in 3‑D | r = a + tb | b ≠ 0. |
| Plane normal form | n·(r−a) = 0 | n = (A,B,C) in general form. |
| Distance point‑plane | d = | (P−A)·n |
| Distance point‑line | d = | (P−A)×b |
| Line–plane intersection | Solve system of line parametric + plane equation. | Usually yields a single point or parallel/identical. |
| Skew lines | Use the formula: distance = | (P₁−P₂)·(d₁×d₂) |
| Projection of u onto v | proj_v(u) = ((u·v)/(v·v)) v | |
| Angle between vectors | cosθ = (u·v)/( | u |
3. Worked Example
Problem
Find:
- The angle between vectors a = ⟨2, -1, 3⟩ and b = ⟨-1, 4, 2⟩.
- The distance from point P(1, 2, 3