Lecture 4 Vectors

1. Scalar Product of Vectors

1.1 Definition

• Let vectors a and b be defined as:

  • a = a1ˆı + a2ˆȷ + a3 ˆk

  • b = b1ˆı + b2ˆȷ + b3ˆk

• The scalar product (dot product) is given by:

  • a · b = a1b1 + a2b2 + a3b3

1.2 Proof of Scalar Product

• Start with a · ˆı:

  • a · ˆı = (a1ˆı + a2ˆȷ + a3 ˆk) · ˆı

  • Using properties:

    • = (a1ˆı) · ˆı + (a2ˆȷ) · ˆı + (a3 ˆk) · ˆı

    • = a1(ˆı · ˆı) + a2(ˆȷ · ˆı) + a3(ˆk · ˆı)

    • = a1(1) + a2(0) + a3(0)

    • Result: a · ˆı = a1

• Similarly,

  • a · ˆȷ = a2

  • a · ˆk = a3

• Components of a recovered through scalar products with basis vectors.

1.3 Magnitude of Vector

• Scalar product of a vector with itself:

  • a · a = |a|²

  • Components lead to:

    • |a|² = a1² + a2² + a3²

    • Therefore, |a| = √(a1² + a2² + a3²)

2. Examples of Scalar Product

2.1 Example 1: Perpendicular Vectors

• Given: a = ˆı − 2ˆȷ + 3ˆk and b = −ˆı + ˆȷ + ˆk

• Calculation:

  • a · b = (1)(−1) + (−2)(1) + (3)(1)

  • = −1 − 2 + 3 = 0

• Conclusion: a and b are perpendicular (90° angle).

2.2 Example 2: Angle Between Vectors

• Given: a = ˆı + ˆȷ − ˆk and b = ˆȷ + ˆk

• Magnitudes:

  • |a| = √(1² + 1² + (−1)²) = √3

  • |b| = √(1² + 1²)

  • = √2

• Calculation of dot product:

  • a · b = (1)(0) + (1)(1) + (−1)(1) = 0

• Angle calculation:

  • cos θ = 0 → θ = 90° (perpendicular).

2.3 Example 3: Calculating Angle with Another Vector

• Given: c = ˆı − ˆȷ − 2ˆk.

• Calculation of magnitudes:

  • |c| = √(1 + 1 + 4) = √6

• Dot product:

  • a · c = 1 − 1 + 2 = 2

• Angle calculation:

  • cos θ = a · c / (|a| |c|) = 2 / (√3 * √6) = √2/3

  • θ = cos⁻¹(√2/3) ≈ 62°.

3. Geometry with Vectors: The Cosine Rule

• For triangle sides A, B, C:

  • Define vectors based on sides: a, b, c = a - b.

• Using magnitudes and scalar products:

  • C² = |c|² = |a - b|² = a · a + b · b - 2 a · b

  • Therefore:

    • C² = A² + B² - 2AB cos θ

• This confirms the cosine rule.

4. Differentiation of Vectors

4.1 Position Vector

• A particle's position in space is given by:

  • r = r(t) = x(t)ˆı + y(t)ˆȷ + z(t)ˆk, where x(t), y(t), z(t) depend on time t.

4.2 Change in Position Vector

• Change from time t to t + Δt:

  • ∆r = r(t + Δt) − r(t) = ∆xˆı + ∆yˆȷ + ∆zˆk

  • This represents the displacement in time Δt.

4.3 Average Change

• Average over time:

  • ∆r/∆t = (∆x/∆t)ˆı + (∆y/∆t)ȷ + (∆z/∆t)ˆk

• Taking the limit as Δt approaches 0 gives:

  • dr/dt = dx/dt ˆı + dy/dt ȷ + dz/dt ˆk

  • This derivative represents the rate of change of the position vector.

4.4 Example of the Derivative

• For r(t) = t²ˆı + (1 - t)ȷ − 3t³ˆk:

  • Derivative calculation:

  • dr/dt = 2tˆı − ˆȷ − 9t²ˆk.