Vector Operations and Problems Lecture

Introduction

• Recap of previous lecture topics
• Several example problems using vector operations

Problem 1: Vector from (0, 2, 0) to (4, 2, 8)

• Vector A Calculation
  • Coordinates conversion: (4-0)i + (2-2)j + (8-0)k
  • A = 4i + 0j + 8k
• Magnitude Calculation
  • ||A|| = sqrt(4^2 + 0^2 + 8^2) = sqrt(80) = 4√5
• Direction Calculation
  • cos(θx) = 4/(4√5) = 1/√5 -> θx = cos⁻¹(1/√5)
  • cos(θy) = 0/(4√5) = 0 -> θy = 90 degrees
  • cos(θz) = 8/(4√5) = 2/√5 -> θz = cos⁻¹(2/√5)

Problem 2: Vector with Rectangular Coordinates (2, 3, 4)

• Vector B Calculation
  • B = 2i + 3j + 4k
• Magnitude Calculation
  • ||B|| = sqrt(2^2 + 3^2 + 4^2) = sqrt(29)
• Direction Calculation
  • cos(θx) = 2/√29 -> θx = cos⁻¹(2/√29)
  • cos(θy) = 3/√29 -> θy = cos⁻¹(3/√29)
  • cos(θz) = 4/√29 -> θz = cos⁻¹(4/√29)

Problem 3: Find a Unit Vector in Direction of C

• Given Vector
  • C = 2i + 3j
• Unit Vector Calculation
  • Magnitude of C = sqrt(2^2 + 3^2) = sqrt(13)
  • C unit = C/||C||
  • C unit = (2/√13)i + (3/√13)j
• Vector with Magnitude of 5
  • D = 5 * C unit
  • D = (10/√13)i + (15/√13)j*

Problem 4: Vector Opposite to a Given Vector

• Given Vector A = 30i + 40j
• Unit Vector Calculation
  • ||A|| = 50
  • A unit = (30/50)i + (40/50)j = 0.6i + 0.8j
• Opposite Vector Calculation
  • D = -10 * A unit
  • D = -6i - 8j*

Problem 5: A + B and Unit Vector in its Direction

• Given Vectors
  • A = 10i + 20j
  • B = 20i + 20j
• Addition and Magnitude Calculation
  • A + B = 30i + 40j
  • ||A + B|| = sqrt(30^2 + 40^2) = 50
• Unit Vector Calculation
  • (A + B) unit = (30/50)i + (40/50)j = 0.6i + 0.8j

Additional Vector Operations

Magnitude and Angle with Z-Axis

• Given Vector
  • A = 3i + 4j + 5k
• Magnitude and Angle Calculation
  • ||A|| = sqrt(3^2 + 4^2 + 5^2) = sqrt(50) = 5√2
  • cos(θz) = 5/(5√2) = 1/√2 -> θz = 45 degrees

Square of Cosines

• Given Relation
  • cos²θx + cos²θy + cos²θz = 1
  • sin²θx + sin²θy + sin²θz = 2

Concept: Addition and Subtraction of Vectors

Resultant Vector

• When two vectors are added/subtracted, the result is a "resultant vector":
  • Let C bar be the resultant vector of A bar and B bar:
  • C = A + B, D = A - B
• Vector Magnitude and Direction
  • Example problems involving calculating resultant vectors, their magnitudes, and directions.

Example Problems

1. Given Displacements
   • Moves North 50m, East 30m, South 10m
   • Resultant displacement calculation
2. Additional Problems
   • Find displacement vectors:
     • North 10m, West 10m, Upwards 10m
     • Calculate resultant displacement

Final Exercises

1. Given velocity vector, find resultant displacement
   • Velocity: 20i + 30j (m/s)
   • Time: 10 seconds
   • Calculate displacement: 200i + 300j, ||s|| = sqrt(200^2+300^2)
2. Given initial and final velocity, find acceleration vector
   • Initial Velocity: 2i - 4k (m/s)
   • Final Velocity: 3j + 8k (m/s)
   • Time: 10 seconds
   • Acceleration Vector: Final - Initial / Time = (-0.2i + 0.3j + 1.2k)
3. Find a component
   • Given magnitude 1, vector = 0.2i + 0.4j + ck
   • Solve for c: c = sqrt(0.8)

Concluding Points

• Key takeaways from the vector operation concepts and problems
• Review of all important steps and correct solutions

Reminder: Complete your worksheets and practice thoroughly before the next class!