Lecture 5 Vectors

1. Vectors in Mechanics

1.1 Velocity Vector

  • Definition: Velocity vector, denoted as v = dr/dt, describes the instantaneous direction of motion and the rate of change of position.

  • Example:

    • If r(t) = (1 + t)ˆı + 2tˆȷ + 3tˆk,

    • For t = -1: r(-1) = -2ˆȷ - 3ˆk

    • For t = 0: r(0) = ˆı

    • For t = 1: r(1) = 2ˆı + 2ˆȷ + 3ˆk

    • For t = 2: r(2) = 3ˆı + 4ˆȷ + 6ˆk

1.2 Calculating Velocity

  • Calculation: v = dr/dt = (0 + 1)ˆı + (2)ˆȷ + (3)ˆk = ˆı + 2ˆȷ + 3ˆk

  • Note: Velocity is independent of time; hence, the tangent vector remains constant along the path indicating motion along a line.

1.3 Non-constant Velocity Example

  • Example: For r(t) = tˆı + (t^2/2)ˆȷ + ˆk,

    • Velocity is v = dr/dt = (1)ˆı + (t)ˆȷ + (0)ˆk = ˆı + tˆȷ.

2. Acceleration Vector

2.1 Definition

  • Acceleration vector is the rate of change of velocity with respect to time: a = dv/dt.

2.2 Example of Acceleration

  • If v = ˆı + tˆȷ,

    • Then, a = dv/dt = ˆȷ.

  • Notation:

    • a = dv/dt = d²r/dt² (second derivative of r with respect to time).

    • Shorthand: v = ˙r, a = ˙v = ¨r, where • indicates a time derivative.

3. Vector Equation of a Line

3.1 Constant Velocity

  • Equation: If a particle moves with constant velocity ˙r = u = u1ˆı + u2ˆȷ + u3ˆk:

    • ẋ = u1 (constant),

    • ẏ = u2 (constant),

    • ż = u3 (constant).

  • Parametric form:

    • x = u1t + C (where C is a constant).

    • If r = r0 at t = 0, then:

      • x = x0 + u1t.

3.2 Vector Representation

  • Complete Equation: r = r0 + tu (vector equation of a line through r0 in the direction of u).

4. Comparison with Standard Line Description

4.1 Description in XY Plane

  • Given initial point r0 and velocity u:

    • x = x0 + u1t

    • y = y0 + u2t

  • From this, we can derive the standard form:

    • y − y0 = (u2/u1)(x − x0) (standard equation of a line).

  • Angles:

    • If u1 = 0, then ϕ = 90°; if u2 = 0, then ϕ = 0°.

5. Example of Line Equations

5.1 Line through a Point

  • Example: For a line through (1, 4, -1) in direction of ˆı - 3ˆk:

    • r = (1 + t)ˆı + 4ˆȷ - (1 + 3t)ˆk.

5.2 Line through Two Points

  • Example: For points (-1, 0, 2) and (2, 1, 3):

    • r0 = -ˆı + 2ˆk, r1 = 2ˆı + ˆȷ + 3ˆk.

    • Direction: r1 - r0 = 3ˆı + ˆȷ + ˆk

    • Line equation: r = -ˆı + 2ˆk + t(3ˆı + ˆȷ + ˆk).

    • Verification: r(0) = r0, r(1) = r1 (correct).

6. Newton’s Laws in Two or Three Dimensions

6.1 Newton’s Second Law

  • Statement: The acceleration produced by a force is proportional to that force.

  • Formula: F = ma (F is the total force vector).

  • Multiple Forces: Total force from several acting forces is given by:

    • F = F1 + F2 + ...

    • In component form, add their components.

6.2 Example of Forces

  • Example: Given forces F1 = ˆı + ˆȷ, F2 = -2ˆı + ˆk, and F3 = ˆȷ - ˆk:

    • Total force: F = (1 - 2 + 0)ˆı + (1 + 0 + 1)ˆȷ + (0 + 1 - 1)ˆk = -ˆı + 2ˆȷ.

6.3 Newton’s First Law

  • Statement: F = 0 implies a = 0; therefore, v is constant (v = u, the initial velocity).

  • Equation of Motion: r(t) = r0 + tu (particle moves with constant velocity in absence of force).