Vectors and Scalars

Scalar Quantities

• Definition: A scalar is any quantity in physics that possesses magnitude only and no inherent direction.

  • Magnitude = numerical value + proper units.

• Common scalar examples (with magnitudes shown in the transcript):

  • Speed – $20\ \text{m/s}$

  • Distance – $10\ \text{m}$

  • Age – $15\ \text{years}$

  • Heat (energy) – $1000\ \text{calories}$

• Conceptual importance:

  • Scalars are added/subtracted by ordinary arithmetic because direction does not matter.

  • Many everyday measurements (time, temperature, mass) are scalars and require no directional notation.

Vector Quantities

• Definition: A vector is any quantity in physics that has both magnitude and direction.

• Standard notation

  • Printed in bold‐face italic with an arrow overhead, e.g. $\vec{v}$, $\vec{F}$.

  • Direction indicated by arrow orientation; magnitude by arrow length (graphically) or by a numerical value + units (algebraically).

• Transcript examples (magnitude, then direction):

  • Velocity – $20\ \text{m/s, North}$

  • Acceleration – $10\ \text{m/s}^2\text{, East}$

  • Force – $5\ \text{N, West}$

• Anatomy of a drawn vector arrow

  • Tail → starting point

  • Head → arrow-point indicating direction

  • Arrow shaft length → proportional to magnitude

• Compass convention for angles/directions (all eight forms listed):

  • $\text{N of E,<br>E of N,<br>S of W,<br>W of S,<br>N of W,<br>W of N,<br>S of E,<br>E of S}$

• Sample directional statements (from Page 6 diagram):

  • Vector A → due North

  • Vector B → $40^{\circ}$ North of West

  • Vector C → $35^{\circ}$ East of South

  • Vector D → $30^{\circ}$ North of East

Vector Addition & Subtraction in One Dimension

• Same direction → add magnitudes

  • Example: Walk $54.5\ \text{m East} + 30\ \text{m East} = 84.5\ \text{m East}$

  • Key observation: Arrow lengths add head-to-tail, overall arrow points East.

• Opposite directions → subtract magnitudes

  • Example: Walk $54.5\ \text{m East} + 30\ \text{m West}$

  • Net: $24.5\ \text{m East}$ (direction of larger component)

Right-Angle (Perpendicular) Vector Addition

• If vectors are perpendicular, construct a right triangle and use Pythagoras:

  • Resultant magnitude: $R = \sqrt{(x\text{‐component})^2 + (y\text{‐component})^2}$

  • Components (legs) lie head-to-toe.

• Transcript example

  • Walk $95\ \text{km East}$ then $55\ \text{km North}$.

  • Magnitude: $R = \sqrt{95^2 + 55^2}\,\text{km} = 109.8\ \text{km}$

  • Direction (angle from East toward North):
    $\theta = \tan^{-1}!\left(\dfrac{55}{95}\right) \approx 30^{\circ}$

  • Complete answer: $109.8\ \text{km, }30^{\circ}\text{ N of E}$

• General trig relations for right triangles (with $x$ horizontal, $y$ vertical):

  • $\cos \theta = \dfrac{x}{R} \;\Rightarrow\; x = R\cos \theta$

  • $\sin \theta = \dfrac{y}{R} \;\Rightarrow\; y = R\sin \theta$

  • $\tan \theta = \dfrac{y}{x}$ (used when $R$ is not yet known)

Naming and Measuring Angles

• A direction label like “North of East” must be paired with an exact angle for complete specification.

• Always state from-compass reference → toward-rotation (e.g.
  $30^{\circ}\ \text{N of E}$ means rotate $30^{\circ}$ from East toward North).

• Angles on full 360° circle often measured counter-clockwise from +x (East) for coordinate calculations.

Worked Example 1 – Multi-Leg Displacement (Bear)

1. Path segments (with direction & magnitude)

   • $35\ \text{m East}$

   • $20\ \text{m North}$

   • $12\ \text{m West}$

   • $6\ \text{m South}$

2. Combine East–West components

   • $35−12 = 23\ \text{m East}$

3. Combine North–South components

   • $20−6 = 14\ \text{m North}$

4. Resultant magnitude
   $R = \sqrt{23^2 + 14^2}\,\text{m} = 26.93\ \text{m}$

5. Direction
   $\theta = \tan^{-1}!\left(\dfrac{14}{23}\right) = 31.3^{\circ}$ (North of East)

6. Final displacement: $26.93\ \text{m, }31.3^{\circ}\text{ N of E}$

Worked Example 2 – Relative Velocity (Boat + River)

1. Boat velocity in still water: $15\ \text{m/s, North}$

2. River current: $8.0\ \text{m/s, West}$

3. Components

   • $v_x = −8.0\ \text{m/s}$ (West negative)

   • $v_y = +15\ \text{m/s}$ (North positive)

4. Resultant speed
   $R = \sqrt{(-8.0)^2 + 15^2}\,\text{m/s} \approx 17\ \text{m/s}$

5. Direction
   $\theta = \tan^{-1}!\left(\dfrac{8.0}{15}\right) = 28.1^{\circ}$ West of North

6. Report: $17\ \text{m/s, }28.1^{\circ}\text{ W of N}$

Rectangular Coordinate System (Reference)

• Positive $x$ → East (right); Positive $y$ → North (up).

• Quadrants numbered counter-clockwise (I: +x,+y; II: −x,+y; III: −x,−y; IV: +x,−y).

• Angles often quoted from +x axis: standard position.

Vector Components – Resolving a Single Force

• Generic formulas (magnitude $F$, angle $\theta$ measured from +x):

  • $F_x = F \cos \theta$

  • $F_y = F \sin \theta$

• Example A (57 N @ $43^{\circ}$ above +x)

  • $F_x = 57\cos 43^{\circ} = 41.69\ \text{N}$

  • $F_y = 57\sin 43^{\circ} = 38.87\ \text{N}$

• Example B (103 N @ $65^{\circ}$ above –x)

  • First convert to standard angle: $180^{\circ}−65^{\circ}=115^{\circ}$

  • $F_x = 103\cos 115^{\circ} = −43.53\ \text{N}$ (negative → West)

  • $F_y = 103\sin 115^{\circ} = 93.35\ \text{N}$ (positive → North)

• Sign convention: Component signs tell quadrant automatically.

General Problem-Solving Strategy for Vectors

• Draw a clear scaled diagram; label all known magnitudes and directions.

• Break each vector into $x$ and $y$ components when NOT colinear.

• Add algebraically: $\Sigma F<em>x$, $\Sigma F</em>y$ (or $\Sigma v<em>x, \Sigma v</em>y$ etc.).

• Compute resultant magnitude with Pythagoras.

• Determine direction using inverse trig; specify with proper compass language.

• Check limiting cases: If one component ≈0, final angle should approach 0° or 90° accordingly.

Conceptual & Real-World Relevance

• Scalars vs. vectors distinction underpins virtually all areas of physics (kinematics, dynamics, electromagnetism).

• Navigation (aviation, maritime) relies on vector addition of velocities (wind/current corrections).

• Engineering uses vector components to determine stresses and forces in multiple dimensions.

• Ethical / safety angle: precise vector calculations prevent accidents in transportation and construction by ensuring correct magnitudes and headings.

Key Equations (Quick Reference)

• Pythagorean Theorem (2-D): $R = \sqrt{a^2 + b^2}$

• Component Formulas: $x = R\cos \theta\,,\; y = R\sin \theta$

• Inverse Tangent for Angle: $\theta = \tan^{-1}!\left(\dfrac{y}{x}\right)$

• Colinear Addition/Subtraction: $R = \pm (A \pm B)$ (sign from dominant direction)

Common Pitfalls & Tips

• Always attach units to magnitudes; an un-unitized number is physically meaningless.

• Draw vectors head-to-tail when adding; tail-to-tail only when constructing components.

• When giving directions, never write just “North of East” – include the numeric angle.

• Use a consistent sign convention; switching mid-solution causes sign errors.

• Approximations: keep at least two extra significant figures in intermediate steps; round final answer responsibly.