Lesson 4: Introduction to Vectors
Estimated time: 35-40 minutes
Learning Objectives
- Define a vector and distinguish it from a scalar
- Write vectors in component form and using unit vectors
- Find the magnitude and direction angle of a vector
- Add vectors and multiply by scalars
- Solve application problems involving force and navigation
What Is a Vector?
Vector — A quantity that has both magnitude (size) and direction. Examples: velocity, force, displacement.
Scalar — A quantity with magnitude only. Examples: temperature, mass, speed.
A vector v from point P(x₁, y₁) to Q(x₂, y₂) has component form: v = ⟨x₂−x₁, y₂−y₁⟩
Magnitude and Direction
Magnitude: |v| = √(a² + b²) for v = ⟨a, b⟩
Direction angle θ: tan θ = b/a (adjust quadrant as needed)
Example 1: Magnitude and Direction
v = ⟨3, 4⟩. |v| = √(9+16) = 5. θ = arctan(4/3) ≈ 53.13°.
Example 2: Component Form from Magnitude and Direction
|v| = 10, θ = 120°. Find component form.
v = ⟨10cos120, 10sin120⟩ = ⟨−5, 5√3⟩ ≈ ⟨−5, 8.66⟩
Vector Operations
Addition: ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩
Scalar multiplication: k⟨a,b⟩ = ⟨ka, kb⟩
Unit vector notation: v = ai + bj
Example 3: Vector Addition
u = ⟨2, 5⟩, v = ⟨−3, 1⟩. u+v = ⟨2+(−3), 5+1⟩ = ⟨−1, 6⟩
Example 4: Unit Vector
Find the unit vector in the direction of v = ⟨3, 4⟩.
u = v/|v| = ⟨3/5, 4/5⟩ = ⟨0.6, 0.8⟩
Applications
Example 5: Resultant Force
Two forces act on an object: F₁ = ⟨40, 30⟩ and F₂ = ⟨−10, 50⟩ (in Newtons). Find the resultant force and its magnitude.
Resultant = ⟨40+(−10), 30+50⟩ = ⟨30, 80⟩
|R| = √(900+6400) = √7300 ≈ 85.44 N
Check Your Understanding
1. Find |⟨−5, 12⟩|.
2. Write ⟨6, −8⟩ as a unit vector times its magnitude.
3. u = ⟨4, −1⟩, v = ⟨2, 3⟩. Find 2u − v.
Key Takeaways
- Vectors have magnitude and direction; scalars have only magnitude.
- Component form: v = ⟨a, b⟩. Magnitude: |v| = √(a²+b²).
- Direction angle: θ = arctan(b/a), adjusted for quadrant.
- Vector addition is component-wise: add the i-components and j-components separately.
- A unit vector has magnitude 1: u = v/|v|.