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: 𝑎 + 𝑏 = 𝑐, 𝑐 - 𝑏 = 𝑎

      • Integration/Differentiation: 𝑑𝑎/𝑑𝑡 or ∫𝑎𝑑𝑡

      • Multiplication/Division: 𝑎 × 𝑏 = 𝑐, 𝑐 ÷ 𝑏 = 𝑎

    • Vector Operations:

      • Addition/Subtraction: 𝑉⃗1 + 𝑉⃗2 = 𝑉⃗3, 𝑉⃗3 − 𝑉⃗2 = 𝑉⃗1

      • Integration/Differentiation: 𝑑𝑉⃗/𝑑𝑡 or ∫𝑉⃗𝑑𝑡

      • Scalar multiplication/division: 𝑘𝑉⃗1 = 𝑉⃗2, 𝑉⃗1 = 𝑉⃗2/𝑘

Cross Product (×)

  • 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(𝜃)

    • Valid only when 𝜃 ≤ 180°

    • Indicates zero cross product for collinear vectors (𝜃 = 0 or π).

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,𝑥𝑖̂ + 𝑉1,𝑦𝑗̂ + 𝑉1,𝑧𝑘̂

    • 𝑉⃗2 = 𝑉2,𝑥𝑖̂ + 𝑉2,𝑦𝑗̂ + 𝑉2,𝑧𝑘̂

  • In a right-handed coordinate system:

    • 𝑖̂ × 𝑗̂ = 𝑘̂, 𝑗̂ × 𝑘̂ = 𝑖̂, 𝑘̂ × 𝑖̂ = 𝑗.

Evaluation of Cross Product via Distribution

  • Distributing terms to find the expression:

    • 𝑉⃗1 × 𝑉⃗2 = (𝑉1,𝑦𝑉2,𝑧 − 𝑉1,𝑧𝑉2,𝑦)𝑖̂ + (𝑉1,𝑧𝑉2,𝑥 − 𝑉1,𝑥𝑉2,𝑧)𝑗̂ + (𝑉1,𝑥𝑉2,𝑦 − 𝑉1,𝑦𝑉2,𝑥)𝑘̂

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

Dot Product (∙)

  • 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 = √(𝑉𝑥² + 𝑉𝑦² + 𝑉𝑧²)

    • Dot Product: 𝑉⃗1∙𝑉⃗2 = 𝑉1,𝑥𝑉2,𝑥 + 𝑉1,𝑦𝑉2,𝑦 + 𝑉1,𝑧𝑉2,𝑧

  • Applications in work and energy calculations; it’s commutative: 𝑉⃗1∙𝑉⃗2 = 𝑉⃗2∙𝑉⃗1.

Dot Product in Magnitudes and Relative Angle

  • When magnitudes |𝑉1| and |𝑉2| are known along with angle 𝜃:

    • 𝑉⃗1∙𝑉⃗2 = |𝑉1||𝑉2|cos(𝜃).

  • Dot product sign:

    • Positive: acute angle (0 ≤ 𝜃 < π/2)

    • Negative: obtuse angle (π/2 < 𝜃 ≤ π)

    • Zero: angle = π/2.

Examples

Cross Product Example
  • Vectors: 𝑉⃗1 = 1𝑖̂ + 2𝑗̂ + 0𝑘̂, 𝑉⃗2 = −2𝑖̂ + 1𝑗̂ + 0𝑘̂

  • Compute using components:

    • 𝑉⃗1 × 𝑉⃗2 = (0 + 0 + 5𝑘̂)

  • Utilizing area and right-hand rule:

    • |𝑉⃗1 × 𝑉⃗2| = 5, direction along +𝑘̂.

Dot Product Example
  • Vectors: 𝑉⃗1 = 1𝑖̂ + 2𝑗̂ + 0𝑘̂, 𝑉⃗2 = −2𝑖̂ + 1𝑗̂ + 0𝑘̂

  • Compute using components:

    • 𝑉⃗1 ∙ 𝑉⃗2 = 0.

  • Utilizing magnitudes and angle:

    • 𝑉⃗1 ∙ 𝑉⃗2 = 0 (angle = 90°).


Conclusion

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

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