4-1 Physics

Module 4-1: Vector Mathematics

Introduction

  • Progress to date has used only basic vector operations: addition, subtraction, scalar multiplication, and division.

  • Question arises: what is the utility in multiplying vectors?

    • Functions as a useful operation with more than one method to form a 'product' of two vectors.

  • Overview of vector operations so far:

    • Scalar Operations:

      • Addition/Subtraction: π‘Ž + 𝑏 = 𝑐, 𝑐 - 𝑏 = π‘Ž

      • Integration/Differentiation: π‘‘π‘Ž/𝑑𝑑 or βˆ«π‘Žπ‘‘π‘‘

      • Multiplication/Division: π‘Ž Γ— 𝑏 = 𝑐, 𝑐 Γ· 𝑏 = π‘Ž

    • Vector Operations:

      • Addition/Subtraction: 𝑉⃗1 + 𝑉⃗2 = 𝑉⃗3, 𝑉⃗3 βˆ’ 𝑉⃗2 = 𝑉⃗1

      • Integration/Differentiation: 𝑑𝑉⃗/𝑑𝑑 or βˆ«π‘‰βƒ—π‘‘π‘‘

      • Scalar multiplication/division: π‘˜π‘‰βƒ—1 = 𝑉⃗2, 𝑉⃗1 = 𝑉⃗2/π‘˜

Cross Product (Γ—)

  • Defined for two non-collinear vectors 𝑉⃗1 and 𝑉⃗2 in a 3D space:

    • These vectors form a plane, not necessarily perpendicular.

    • Cross product gives a vector 𝑉⃗3 that is perpendicular to the plane formed by 𝑉⃗1 and 𝑉⃗2.

    • Magnitude equals the area of the parallelogram defined by 𝑉⃗1 and 𝑉⃗2.

  • Important notes on collinearity:

    • Collinear vectors do not define a plane, making the cross product zero in such cases.

Magnitude of Cross Product

  • Area of the parallelogram can be calculated using:

    • 𝑉3 = |𝑉1||𝑉2| sin(πœƒ)

    • Valid only when πœƒ ≀ 180Β°

    • Indicates zero cross product for collinear vectors (πœƒ = 0 or Ο€).

Direction of Cross Product

  • Determined using the right-hand rule:

    • Point fingers of the right hand along 𝑉⃗1 and curl towards 𝑉⃗2.

    • Thumb direction indicates the cross product direction.

  • Ordering matters; reversing vectors in cross product reverses the resultant vector's direction.

Cross Product in Component Form

  • Vectors defined by components:

    • 𝑉⃗1 = 𝑉1,π‘₯𝑖̂ + 𝑉1,𝑦𝑗̂ + 𝑉1,π‘§π‘˜Μ‚

    • 𝑉⃗2 = 𝑉2,π‘₯𝑖̂ + 𝑉2,𝑦𝑗̂ + 𝑉2,π‘§π‘˜Μ‚

  • In a right-handed coordinate system:

    • 𝑖̂ Γ— 𝑗̂ = π‘˜Μ‚, 𝑗̂ Γ— π‘˜Μ‚ = 𝑖̂, π‘˜Μ‚ Γ— 𝑖̂ = 𝑗.

Evaluation of Cross Product via Distribution

  • Distributing terms to find the expression:

    • 𝑉⃗1 Γ— 𝑉⃗2 = (𝑉1,𝑦𝑉2,𝑧 βˆ’ 𝑉1,𝑧𝑉2,𝑦)𝑖̂ + (𝑉1,𝑧𝑉2,π‘₯ βˆ’ 𝑉1,π‘₯𝑉2,𝑧)𝑗̂ + (𝑉1,π‘₯𝑉2,𝑦 βˆ’ 𝑉1,𝑦𝑉2,π‘₯)π‘˜Μ‚

  • Application primarily uses magnitude from area formulas and the right-hand rule.

Dot Product (βˆ™)

  • Cross product yields a vector; dot product produces a scalar (size only, no direction).

  • Scalar defined by coordinate independence. Changing coordinates does not change the scalar description.

Dot Product from Vector Components

  • The dot product's essence:

    • Magnitude Calculation: 𝑉 = √(𝑉π‘₯Β² + 𝑉𝑦² + 𝑉𝑧²)

    • Dot Product: 𝑉⃗1βˆ™π‘‰βƒ—2 = 𝑉1,π‘₯𝑉2,π‘₯ + 𝑉1,𝑦𝑉2,𝑦 + 𝑉1,𝑧𝑉2,𝑧

  • Applications in work and energy calculations; it’s commutative: 𝑉⃗1βˆ™π‘‰βƒ—2 = 𝑉⃗2βˆ™π‘‰βƒ—1.

Dot Product in Magnitudes and Relative Angle

  • When magnitudes |𝑉1| and |𝑉2| are known along with angle πœƒ:

    • 𝑉⃗1βˆ™π‘‰βƒ—2 = |𝑉1||𝑉2|cos(πœƒ).

  • Dot product sign:

    • Positive: acute angle (0 ≀ πœƒ < Ο€/2)

    • Negative: obtuse angle (Ο€/2 < πœƒ ≀ Ο€)

    • Zero: angle = Ο€/2.

Examples

Cross Product Example
  • Vectors: 𝑉⃗1 = 1𝑖̂ + 2𝑗̂ + 0π‘˜Μ‚, 𝑉⃗2 = βˆ’2𝑖̂ + 1𝑗̂ + 0π‘˜Μ‚

  • Compute using components:

    • 𝑉⃗1 Γ— 𝑉⃗2 = (0 + 0 + 5π‘˜Μ‚)

  • Utilizing area and right-hand rule:

    • |𝑉⃗1 Γ— 𝑉⃗2| = 5, direction along +π‘˜Μ‚.

Dot Product Example
  • Vectors: 𝑉⃗1 = 1𝑖̂ + 2𝑗̂ + 0π‘˜Μ‚, 𝑉⃗2 = βˆ’2𝑖̂ + 1𝑗̂ + 0π‘˜Μ‚

  • Compute using components:

    • 𝑉⃗1 βˆ™ 𝑉⃗2 = 0.

  • Utilizing magnitudes and angle:

    • 𝑉⃗1 βˆ™ 𝑉⃗2 = 0 (angle = 90Β°).


Conclusion

  • Understanding both cross and dot products is crucial in physics for analyzing motion, forces, torque, and energy.

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