Overview
This lecture explains how to represent vectors with components, convert between notations, perform vector addition/subtraction, and compute vector magnitudes using component forms.
Component Representation of Vectors
- Vectors can be split into three unit vectors: i (x-direction), j (y-direction), and k (z-direction).
- Any vector in 3D can be formed by summing scaled versions of i, j, and k.
- The coefficients of i, j, and k (called components) define the vector.
- The standard component notation is angle brackets, e.g., <v₁, v₂, v₃>, representing (x, y, z) components.
- Other notations exist (round brackets, square brackets, etc.), but meaning depends on context.
Translating Between Graphical and Component Notations
- A vector moving 2 steps right is <2,0> (x=2, y=0).
- A vector 1 right and 1 up is <1,1>.
- A vector 1 right and 3 down is <1,-3>.
- A vector 2 left and 2 up is <-2,2>.
Vector Addition and Subtraction with Components
- To add vectors, add corresponding components: <x₁, y₁> + <x₂, y₂> = <x₁+x₂, y₁+y₂>.
- Example: <2,0> + <1,-3> = <3,-3>.
- To subtract, subtract components: <x₁, y₁> - <x₂, y₂> = <x₁-x₂, y₁-y₂>.
- Example: <-2,2> - <1,-3> = <-3,5>.
Calculating Magnitude of a Vector
- For vector <v₁, v₂>, magnitude = √(v₁² + v₂²).
- For <v₁, v₂, v₃>, magnitude = √(v₁² + v₂² + v₃²).
- Example: Magnitude of <2, -3> = √(2² + (-3)²) = √(4+9) = √13.
Key Terms & Definitions
- Vector — A quantity with both magnitude and direction.
- Component — The value of a vector in the direction of each axis (x, y, z).
- Unit Vector (i, j, k) — Vectors of length 1 along the x, y, z axes.
- Magnitude — The length of a vector, calculated using the Pythagorean theorem.
Action Items / Next Steps
- Practice converting vectors between graphical and component forms.
- Try adding and subtracting vectors using components.
- Calculate magnitudes for several example vectors in component form.