PHYS1702 W1L L3

Vector Basics: Magnitude & Direction

• A vector is any quantity that possesses both magnitude and direction.
  • Common visual: an arrow.
  • Length of arrow ∝ magnitude.
  • Orientation of arrow sets its direction relative to chosen axes (x and y in 2-D).
• Formal “polar” description
  • Specify two pieces of data:
  1. Magnitude C (often just the numerical size, no sign)
  2. Direction angle \theta measured from the reference axes
  • E.g., “Vector \mathbf C has magnitude 5\,\text{units} at 20^\circ above the x-axis.”

Component (Cartesian) Representation

• Most useful form in practice: break the vector into projections along coordinate axes.
  • In 2-D: \mathbf C \;\longrightarrow\;(Cx,\,Cy)
  • In 3-D: \mathbf C \;\longrightarrow\;(Cx,\,Cy,\,C_z)
• Components describe how far the vector moves purely in each independent direction.
• Preferred because algebra on vectors (add, subtract, scale) becomes straightforward “number-by-number.”

Converting Between Polar & Cartesian Forms

• Given Cx and Cy, recover magnitude via Pythagoras:
  • C = \sqrt{Cx^2 + Cy^2}
• Recover direction angle using tangent definition in the right triangle:
  • \tan\theta = \tfrac{Cy}{Cx} (opposite ÷ adjacent)
  • Therefore \theta = \operatorname{arctan}\bigl(\tfrac{Cy}{Cx}\bigr)
  • Same as “\tan^{-1}” on scientific calculators, or Arctan[] in Mathematica.
• Reverse process: starting with magnitude C and angle \theta
  • C_x = C \cos\theta
  • C_y = C \sin\theta

Vector Operations in Component Form

1. Addition

• If \mathbf A = (Ax, Ay) and \mathbf B = (Bx, By),
  • \mathbf A + \mathbf B = (Ax + Bx,\; Ay + By).
• Intuitive rule: add like coordinates together.

2. Subtraction

• \mathbf A - \mathbf B = (Ax - Bx,\; Ay - By).

3. Scalar Multiplication

• Multiply each component by the scalar constant k.
  • Example: 2\mathbf A = (2Ax,\; 2Ay).
  • Example: -3\mathbf A = (-3Ax,\; -3Ay).
• Works analogously in 3-D.

Graphical Interpretation: Head-to-Tail Addition

• Procedure
  1. Draw \mathbf A as an arrow.
  2. Translate (without rotating) \mathbf B so its tail touches the head of \mathbf A.
  3. The arrow from the start of \mathbf A to the end of \mathbf B represents \mathbf C = \mathbf A + \mathbf B.
• Why it matches component addition
  • Introduce coordinate axes and drop perpendiculars to form right triangles.
  • Total horizontal span = Ax + Bx → matches C_x.
  • Total vertical span = Ay + By → matches C_y.
  • Hence geometric and algebraic definitions are identical.
• Same logic extends to vector subtraction (place -\mathbf B