3.2 Coordinate System and Vector Components — Key Points

Coordinate systems

• A coordinate system is an artificial grid imposed on a problem to make quantitative measurements easier. We usually use Cartesian coordinates: two perpendicular axes with an origin where they cross.
• Label axes clearly (e.g., +x to the right, +y up).

Component vectors and vector resolution

• Any vector a can be decomposed into two perpendicular components parallel to the axes: a = ax + ay.
• The x component vector ax lies along the x-axis; the y component vector ay lies along the y-axis.
• Components are scalars with sign: ax and ay carry magnitude and direction information.
• The parallelogram (or head-to-tail) rule shows that a is the sum of its components.

How to determine components from magnitude and angle

• If the vector a has magnitude |a| and is at angle θ from the +x axis (above the horizontal):
  $a<em>x = |a| \cos \theta, \quad a</em>y = |a| \sin \theta.$
• Signs reflect direction: positive components point toward the positive axis direction.
• If the angle is measured from the vertical (the y-axis), the roles of sine and cosine swap (examples below):
  $a<em>x = |a| \sin \theta, \quad a</em>y = -|a| \cos \theta.$
• The magnitude and angle from components:
  $|a| = \sqrt{a<em>x^2 + a</em>y^2}, \quad \theta = \tan^{-1}\left(\frac{a<em>y}{a</em>x}\right).$
• Important: always define the angle from a diagram; the above formulas depend on that definition and may change if the angle origin changes.

Determine components from a given vector direction (examples)

• Example: magnitude 6.0 m/s^2 at 30° below the negative x-axis (i.e., in the third quadrant direction along −x and −y with 30° down from −x):
  $a<em>x = -6.0 \cos 30^{\circ} \approx -5.2,$$a</em>y = -6.0 \sin 30^{\circ} = -3.0.$
• Example: vector with components ax = +3 m, ay = +2 m corresponds to a vector in the first quadrant; signs indicate direction.

Inverse problem: from components to magnitude and direction

• Magnitude from components:
  $|a| = \sqrt{a<em>x^2 + a</em>y^2}.$
• Direction from components:
  $\theta = \tan^{-1}\left(\frac{a<em>y}{a</em>x}\right)$
• If signs differ (different quadrant), adjust θ accordingly based on the signs of ax and ay.
• If angle is defined relative to vertical, use the appropriate form (see above).

Special cases: vectors with negative components and quadrant awareness

• If a vector points down or left, the corresponding components are negative (e.g., ay < 0 for downward, ax < 0 for left).
• The magnitude is unaffected by signs, but direction is.
• When the vector lies in unusual quadrants, always refer to a diagram to identify the correct signs and angle.

Vector addition and subtraction using components (algebraic addition)

• If c = a + b, then
  $c<em>x = a</em>x + b<em>x, \quad c</em>y = a<em>y + b</em>y.$
• Subtraction: if d = p − q, then
  $d<em>x = p</em>x - q<em>x, \quad d</em>y = p<em>y - q</em>y.$
• Scalar multiplication: for scalar t,
  $(t\mathbf{a})<em>x = t a</em>x, \quad (t\mathbf{a})<em>y = t a</em>y.$
• Vector equations encode two scalar equations (one for each axis).

Example: displacement by algebraic (component) addition

• Bird displacement: 100 m east, then 200 m NW (45° north of west).
  • Components: Ax = +100 m, Ay = 0; Bx = -200 cos 45° ≈ -141 m, By = +200 sin 45° ≈ +141 m.
  • Net: Cx = Ax + Bx ≈ -41 m; Cy = Ay + By ≈ +141 m.
  • Magnitude: |C| ≈ \sqrt{(-41)^2 + 141^2} ≈ 147\,\text{m}.
  • Direction: θ ≈ arctan(|141|/|41|) ≈ 74°, i.e., 74° north of west.

Tilted coordinate systems (tilted axes)

• Sometimes x' axis is along a slope; decompose vectors into components along the tilted axes.
• In a tilted frame, components are still projections:
  $c<em>{x'} = c \cos \theta', \quad c</em>{y'} = c \sin \theta'$
  where θ' is the angle with respect to the tilted x' axis.
• If the angle is measured from the vertical in a tilted frame, formulas adjust similarly (use the appropriate projections).

Quick reference tips

• Components are the projections of a vector onto the axes; the numbers carry sign to indicate direction.
• Always define the angle from a diagram before applying formulas.
• Use algebraic addition for combining vectors via their components; this is often simpler than graphical addition.
• Magnitude is found via the Pythagorean relation; direction via arctangent with quadrant awareness.