Vector — Quantity with Magnitude and Direction

A vector is a mathematical entity that has both magnitude (size) and direction. In science and engineering, vectors are indispensable for describing physical quantities where orientation matters, such as force, velocity, and displacement. Unlike scalars—which are fully described by a single value (e.g., mass, temperature)—vectors require both a value and a direction.

Key Concepts

• Vector: A quantity with both magnitude and direction, typically represented algebraically as ordered pairs (2D), triples (3D), or n-tuples (nD), and graphically as arrows.
• Scalar: A quantity with only magnitude, no direction (e.g., temperature, mass).
• Magnitude: The length or size of the vector.
• Direction: The orientation of the vector, often described by an angle with respect to a reference axis.
• Components: The projections of a vector along coordinate axes; in 2D: x and y, in 3D: x, y, z.
• Unit Vector: A vector of magnitude 1, indicating direction only.
• Resultant Vector: The sum or combined effect of two or more vectors.
• Displacement: A vector describing the straight-line distance and direction from an initial to a final position.

How Are Vectors Used?

Vectors are essential tools in numerous fields:

• Physics: To describe forces, velocities, accelerations, momenta, and fields.
• Engineering: For analyzing structures, calculating stresses, and controlling robotics.
• Navigation & Aviation: To plot course headings, wind correction, and wayfinding.
• Earth Sciences: To represent tectonic plate motion, earthquake slip, and flow directions.
• Computer Graphics: For rendering motion, lighting, and spatial transformations.

Real-World Example: Tectonic Plate Motion

On tectonic maps, arrows (vectors) indicate the motion of plates. The length of each arrow reflects the speed (e.g., mm/year), and its orientation shows direction. Scientists use these vectors to analyze plate boundaries, strain accumulation, and seismic risk.

Vector vs. Scalar: Quick Reference

How Are Vectors Represented?

1. Geometric (Arrow) Form

Vectors are commonly drawn as arrows. The tail marks the starting point; the tip points in the intended direction. The arrow’s length is proportional to magnitude.

2. Component (Cartesian) Form

Vectors can be written as ordered pairs or triples:

• In 2D: v = ⟨x, y⟩
• In 3D: v = ⟨x, y, z⟩

If a vector starts at (x₀, y₀) and ends at (x₁, y₁):

v = ⟨x₁ − x₀, y₁ − y₀⟩

3. Unit Vector Notation

• 2D: v = a·i + b·j
• 3D: v = a·i + b·j + c·k

Where i, j, and k are unit vectors along x, y, and z axes, respectively.

Magnitude and Direction of a Vector

Given v = ⟨x, y⟩:

• Magnitude:
  |v| = √(x² + y²)
• Direction (angle θ):
  θ = arctan(y / x) (use atan2(y, x) for correct quadrant)

For 3D, |v| = √(x² + y² + z²).

Worked Example

From P(1, 1) to Q(5, 3):

• Components: ⟨5−1, 3−1⟩ = ⟨4, 2⟩
• Magnitude: √(4² + 2²) = √20 ≈ 4.47
• Direction: θ = arctan(2/4) ≈ 26.57°

Breaking Vectors into Components

A vector with magnitude v and angle θ:

• x-component: vₓ = v·cos(θ)
• y-component: v_y = v·sin(θ)

Example:
Wind blows at 50 knots, 30° east of north:

• East component: 50·sin(30°) = 25 knots
• North component: 50·cos(30°) ≈ 43.3 knots

Vector Operations

Addition

If a = ⟨aₓ, a_y⟩, b = ⟨bₓ, b_y⟩:

a + b = ⟨aₓ + bₓ, a_y + b_y⟩

Graphically: Place the tail of the second vector at the tip of the first (tip-to-tail).

Scalar Multiplication

Multiplying by k:

k·v = ⟨k·vₓ, k·v_y⟩

If k < 0, the vector reverses direction.

Real-World Use Cases

• Tectonic Plate Motion: Vectors show speed and direction of plate movement.
• Landslide Forces: Gravity vector split into slope-parallel and normal components.
• Navigation & GPS: Displacement vectors determine shortest path and heading.
• Physics & Engineering: Vectors underpin Newton’s laws, projectile motion, and torque.
• Aviation: Pilots use vectors for wind correction and route planning.

Practice Problems

1. Find the magnitude and direction of the vector from A(2,2) to B(7,6).

   • Components: ⟨7−2, 6−2⟩ = ⟨5, 4⟩
   • Magnitude: √(5² + 4²) = √41 ≈ 6.4
   • Direction: θ = arctan(4/5) ≈ 38.7°

2. A plane flies 200 km east, then 150 km north. Find the resultant displacement vector’s magnitude and direction.

   • Components: ⟨200, 150⟩
   • Magnitude: √(200² + 150²) = √(40000 + 22500) = √62500 = 250 km
   • Direction: θ = arctan(150/200) ≈ 36.9° north of east

Summary

Vectors are fundamental quantities in mathematics, physics, engineering, and navigation. Their power lies in representing both magnitude and direction, allowing precise modeling of real-world phenomena from forces and velocities to motion and navigation. Mastery of vector concepts enables effective analysis and problem-solving in countless scientific and technical domains.