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Introduction to Vectors and Operations

Jul 14, 2024

Introduction to Vectors and Operations

Basic Definitions

  • Vector: Quantity with both direction and magnitude.
    • Represented by an arrow, with length and direction.
    • Computed using a coordinate system (x-y plane, x-y-z space).

Notation and Components

  • Vector A: Notation as A (with an arrow on top).
  • Unit Vectors: Defined along coordinate axes.
    • i-hat: Points along the x-axis with length 1.
    • j-hat: Points along the y-axis with length 1.
    • k-hat: Points along the z-axis with length 1.
  • Components: Represented as [a1, a2, a3] (angular brackets).
  • Length of a Vector: Denoted as |A|, represents the scalar quantity (magnitude).

Important Concepts

  • Direction (dir A): Obtained by scaling the vector to unit length.
  • Vector with Two Points P and Q: Vector from P to Q is denoted PQ.

Calculation of Vector Length

  • Example: Vector <3,2,1>.
    • Uses Pythagorean theorem to find length.
    • Length of A = √(3^2 + 2^2 + 1^2) = √14.

Adding and Scaling Vectors

  • Vector Addition: Combine two vectors:
    • Geometric method: Place vectors head to tail.
    • Numerical method: Add corresponding components.
    • Example: A = [a1, a2, a3], B = [b1, b2, b3]; A + B = [a1 + b1, a2 + b2, a3 + b3].
  • Scalar Multiplication: Multiply vector by a number (scalar).
    • Example: 2A = A scaled by 2.
    • Can be written as 3i 2j k meaning scaling unit vectors by components.

Dot Product

  • Definition: Multiply components of two vectors and sum.
    • A • B = a1b1 + a2b2 + a3b3.
    • Result is a scalar.
  • Geometric Interpretation:
    • A • B = |A| |B| cos(θ) (θ is the angle between A and B).

Applications of Dot Product

  1. Calculating Lengths and Angles:
  • Example: P (1,0,0), Q (0,1,0), R (0,0,2).
    • Calculate angle using dot product and lengths.
    • Cosine θ = PQ • PR / (|PQ| |PR|) = 1/√10.
  1. Detecting Perpendicularity (Orthogonality):
  • Two vectors are perpendicular if A • B = 0.
    • Example equation of a plane: x + 2y + 3z = 0 interpreted as OP • A = 0 (plane perpendicular to vector A).

Summary

  • Dot product helps in understanding lengths and angles.
  • Vectors and their operations can be used to represent and simplify geometric problems.
  • Vectors in more than three dimensions follow the same principles but are harder to visualize.
  • Future Topics: Cross products, further operations with vectors in geometry.

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