Rucete ✏ AP Precalculus In a Nutshell
11. Vectors
This chapter introduces vectors in both algebraic and geometric forms. You’ll learn to represent motion with vectors, perform vector operations, compute dot products, analyze direction, and use vectors in modeling physical forces.
- Vector Basics
- A vector has both magnitude and direction, unlike scalars.
- Written as ⟨x, y⟩; represented by arrows.
- Vectors can originate from any point.
- Magnitude = √(x² + y²); direction = angle from x-axis.
- Between two points A to B: v = ⟨x₂ − x₁, y₂ − y₁⟩
- Equality and Negation
- Vectors are equal if their components match exactly.
- Negative vector has same magnitude, opposite direction: −v = ⟨−a, −b⟩
- Vector Addition and Subtraction
- Graphically: head-to-tail method
- Algebraically:
- ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩
- ⟨a, b⟩ − ⟨c, d⟩ = ⟨a − c, b − d⟩
- Subtraction = addition of negative vector
- Scalar Multiplication
- Multiply each component: k⟨a, b⟩ = ⟨ka, kb⟩
- k > 0: same direction
- k < 0: opposite direction
- k = 0: zero vector
- Dot Product and Angle Between Vectors
- u ⋅ v = a₁b₁ + a₂b₂
- u ⋅ v = |u||v|cosθ
- u ⋅ v = 0 ⇔ perpendicular
- u ⋅ v > 0 ⇔ acute angle
- u ⋅ v < 0 ⇔ obtuse angle
- Word Problems with Vectors
- Model forces like wind or tension
- Use law of cosines and law of sines with vector triangles
- Example:
- Forces: 25 lb and 60 lb at 30° → use law of cosines
- Find angle using law of sines
- Component Form and Unit Vectors
- ⟨a, b⟩ = a𝑖 + b𝑗, where 𝑖 = ⟨1, 0⟩, 𝑗 = ⟨0, 1⟩
- Unit vector: magnitude = 1
- û = v / |v|
- Direction and magnitude form: v = |v|⟨cosθ, sinθ⟩
- Vector-Valued Functions
- r(t) = ⟨x(t), y(t)⟩ models motion
- Velocity = derivative of position
- Speed = magnitude of velocity
- Example:
- r(t) = ⟨t² − t, t² + t⟩ → r(2) = ⟨2, 6⟩, distance = √40
- Velocity = r′(t); speed = √[(dx/dt)² + (dy/dt)²]
In a Nutshell
Vectors describe quantities with direction and magnitude and are essential in physics and motion modeling. You can add, subtract, scale, and dot vectors both algebraically and graphically. Using vector-valued functions, it's possible to model real-world systems like particle motion, forces, and direction—all fundamental for calculus and science.