Vectors in the Plane

What is a Vector?

Vector: Quantity with both magnitude (size) and direction

Examples:

• Displacement: 5 miles northeast
• Velocity: 60 mph due east
• Force: 10 N downward

Contrast with scalar: Magnitude only (temperature, mass, time)

Notation:

• Bold: v
• Arrow: v⃗
• Component form: ⟨a, b⟩ or ⟨a, b⟩

Geometric Representation

Arrow from initial point to terminal point

Key features:

• Length = magnitude
• Direction = angle or slope
• Position doesn't matter (can translate)

Equal vectors: Same magnitude and direction (can be at different positions)

Example: Vector Representation

Arrow from (1, 2) to (4, 5)

This is the same vector as arrow from (0, 0) to (3, 3)

Both have same displacement: 3 right, 3 up

Component Form

Component form: ⟨a, b⟩ or ai + bj

Where:

• a = horizontal component (x-direction)
• b = vertical component (y-direction)

From points: If vector from P₁(x₁, y₁) to P₂(x₂, y₂):

v⃗ = ⟨x₂ - x₁, y₂ - y₁⟩

Example 1: Find Component Form

Vector from A(2, 3) to B(5, 7)

Calculate:

v⃗ = ⟨5 - 2, 7 - 3⟩
  = ⟨3, 4⟩

Answer: ⟨3, 4⟩

Example 2: Starting at Origin

Vector from origin to (6, -2)

Component form: ⟨6, -2⟩

Position vector

Magnitude of a Vector

Magnitude (length): |v⃗| or ‖v⃗‖

Formula: |⟨a, b⟩| = √(a² + b²)

Derived from Pythagorean theorem

Example 1: Find Magnitude

v⃗ = ⟨3, 4⟩

Calculate:

|v⃗| = √(3² + 4²)
    = √(9 + 16)
    = √25
    = 5

Answer: 5

Example 2: From Points

Vector from (1, 2) to (4, 6)

Component form: ⟨3, 4⟩

Magnitude: √(9 + 16) = 5

Direction of a Vector

Direction angle: Angle θ from positive x-axis (counterclockwise)

Formula: tan(θ) = b/a for vector ⟨a, b⟩

Or: θ = tan⁻¹(b/a)

Note: Adjust for quadrant!

Example 1: Find Direction

v⃗ = ⟨3, 3⟩

Calculate:

tan(θ) = 3/3 = 1
θ = tan⁻¹(1) = 45°

Answer: 45° (northeast direction)

Example 2: Quadrant Adjustment

v⃗ = ⟨-4, 3⟩

Calculate:

tan⁻¹(3/-4) ≈ -36.9°

But vector in Quadrant II

Correct angle: 180° - 36.9° = 143.1°

Unit Vectors

Unit vector: Vector with magnitude 1

Standard unit vectors:

• i = ⟨1, 0⟩ (horizontal)
• j = ⟨0, 1⟩ (vertical)

Any vector: v⃗ = ⟨a, b⟩ = ai + bj

Unit vector in direction of v⃗:

u⃗ = v⃗/|v⃗| = (1/|v⃗|)⟨a, b⟩

Example: Find Unit Vector

v⃗ = ⟨3, 4⟩

Magnitude: |v⃗| = 5

Unit vector:

u⃗ = ⟨3, 4⟩/5
  = ⟨3/5, 4/5⟩

Check: |⟨3/5, 4/5⟩| = √(9/25 + 16/25) = √(25/25) = 1 ✓

Vector Addition

Add component-wise

⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩

Geometric: "Tip-to-tail" method or parallelogram rule

Example 1: Add Vectors

u⃗ = ⟨2, 3⟩, v⃗ = ⟨4, -1⟩

Calculate:

u⃗ + v⃗ = ⟨2 + 4, 3 + (-1)⟩
      = ⟨6, 2⟩

Answer: ⟨6, 2⟩

Example 2: Geometric Interpretation

Walk 3 blocks east (⟨3, 0⟩), then 4 blocks north (⟨0, 4⟩)

Total displacement:

⟨3, 0⟩ + ⟨0, 4⟩ = ⟨3, 4⟩

Net: 3 east, 4 north

Vector Subtraction

Subtract component-wise

⟨a, b⟩ - ⟨c, d⟩ = ⟨a - c, b - d⟩

Geometric: Vector from second to first

Example: Subtract Vectors

v⃗ = ⟨5, 3⟩, w⃗ = ⟨2, 1⟩

Calculate:

v⃗ - w⃗ = ⟨5 - 2, 3 - 1⟩
      = ⟨3, 2⟩

Answer: ⟨3, 2⟩

Scalar Multiplication

Multiply each component by scalar

k⟨a, b⟩ = ⟨ka, kb⟩

Effects:

• |k| > 1: stretches
• 0 < |k| < 1: shrinks
• k < 0: reverses direction

Example 1: Scalar Multiplication

v⃗ = ⟨2, 3⟩

3v⃗ = ⟨6, 9⟩ (stretched by 3)

-2v⃗ = ⟨-4, -6⟩ (reversed and doubled)

½v⃗ = ⟨1, 1.5⟩ (halved)

Example 2: Application

Velocity v⃗ = ⟨30, 40⟩ km/h

After 2 hours:

Displacement = 2v⃗ = ⟨60, 80⟩ km

Zero Vector

Zero vector: ⟨0, 0⟩ or 0⃗

Properties:

• Magnitude = 0
• No specific direction
• Identity for addition: v⃗ + 0⃗ = v⃗

Parallel Vectors

Parallel: One is scalar multiple of other

v⃗ ∥ w⃗ if v⃗ = kw⃗ for some scalar k

Example: Check if Parallel

u⃗ = ⟨2, 4⟩, v⃗ = ⟨3, 6⟩

Check:

v⃗ = (3/2)u⃗ = ⟨3, 6⟩

Yes, parallel (v⃗ = 1.5u⃗)

Vector from Magnitude and Direction

Given |v⃗| and angle θ:

v⃗ = ⟨|v⃗|cos(θ), |v⃗|sin(θ)⟩

Example: Construct Vector

Magnitude 10, angle 60°

Calculate:

v⃗ = ⟨10cos(60°), 10sin(60°)⟩
  = ⟨10(0.5), 10(√3/2)⟩
  = ⟨5, 5√3⟩

Answer: ⟨5, 5√3⟩

Linear Combinations

Express vector as combination of i and j

v⃗ = ai + bj = ⟨a, b⟩

Example: Linear Combination

v⃗ = 3i - 2j

Component form: ⟨3, -2⟩

Magnitude: √(9 + 4) = √13

Real-World Applications

Physics: Force, velocity, acceleration

Engineering: Stress, strain, displacement

Navigation: Wind velocity, aircraft heading

Computer graphics: Object movement, transformations

Robotics: Position, orientation

Example: Force Vector

Pull box with 50 N force at 30° above horizontal

Force vector:

F⃗ = ⟨50cos(30°), 50sin(30°)⟩
  = ⟨50(√3/2), 50(0.5)⟩
  = ⟨25√3, 25⟩ N

Horizontal component: ≈ 43.3 N Vertical component: 25 N

Example: Navigation

Plane velocity: ⟨200, 0⟩ km/h east Wind velocity: ⟨0, 50⟩ km/h north

Resultant velocity:

v⃗ = ⟨200, 0⟩ + ⟨0, 50⟩ = ⟨200, 50⟩

Speed: |v⃗| = √(40000 + 2500) ≈ 206 km/h

Direction: tan⁻¹(50/200) ≈ 14° north of east

Practice