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.