AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.8B — Vectors
Notes — Addition, Subtraction, and Scalar Multiplication
💡 Learning Objectives (4.8.A Part 2)
By the end of this lesson you will be able to:
- Add and subtract vectors algebraically and geometrically
- Multiply a vector by a scalar and interpret the result
- Recognize the parallelogram and head-to-tail rules for addition
- Solve problems involving vector combinations of forces or velocities
1. Vector Addition — Algebraically
To add two vectors, simply add their corresponding components:
⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩
📘 Example — Addition
- ⟨3, 5⟩ + ⟨−1, 2⟩ = ⟨2, 7⟩
- ⟨4, −2⟩ + ⟨−4, 6⟩ = ⟨0, 4⟩
2. Vector Addition — Geometrically (Two Pictures)
There are two equivalent geometric pictures for adding vectors:
- Head-to-tail rule: Place the tail of v at the head of u. The sum u + v is the vector from u's tail to v's head.
- Parallelogram rule: Place u and v with their tails at the same point. Complete the parallelogram. The sum is the diagonal from the shared tail.
Both rules give the same answer; pick whichever is easier for the problem at hand.
3. The Zero Vector and Additive Inverse
The ZERO VECTOR is 0 = ⟨0, 0⟩. It has no magnitude and no defined direction. Adding 0 to any vector leaves it unchanged.
The NEGATIVE of a vector v = ⟨a, b⟩ is −v = ⟨−a, −b⟩. Geometrically, −v points in the OPPOSITE direction with the same magnitude.
4. Vector Subtraction
Subtraction is just adding the negative:
u − v = u + (−v) = ⟨u₁ − v₁, u₂ − v₂⟩
Geometrically: if both vectors start at the same point, u − v is the vector from the HEAD of v to the HEAD of u (the ‘shortcut’ from v's tip to u's tip).
📘 Example — Subtraction
- ⟨5, 3⟩ − ⟨2, 7⟩ = ⟨3, −4⟩
- ⟨−1, 4⟩ − ⟨2, 4⟩ = ⟨−3, 0⟩
5. Scalar Multiplication
Multiplying a vector by a number (a scalar) STRETCHES, SHRINKS, or REVERSES it without changing its line of action:
k · ⟨a, b⟩ = ⟨k·a, k·b⟩
- If k > 0: same direction, magnitude multiplied by k
- If k < 0: opposite direction, magnitude multiplied by |k|
- If k = 0: result is the zero vector
- |k · v| = |k| · |v|
📘 Example — Scalar multiplication
- 3 · ⟨2, −1⟩ = ⟨6, −3⟩
- −2 · ⟨4, 5⟩ = ⟨−8, −10⟩
- (1/2) · ⟨10, −6⟩ = ⟨5, −3⟩
6. Properties of Vector Operations
💡 Key Properties
- Commutativity of addition: u + v = v + u
- Associativity: (u + v) + w = u + (v + w)
- Distributivity: k(u + v) = k·u + k·v
- Distributivity over scalars: (k + m)·v = k·v + m·v
- Zero vector: v + 0 = v
- Additive inverse: v + (−v) = 0
7. Linear Combinations
A LINEAR COMBINATION of vectors u and v is any vector of the form a·u + b·v for scalars a, b. Every vector in the plane can be written as a linear combination of two non-parallel vectors — this is one reason i and j are so useful as a ‘basis.’
📘 Example — Linear combination
- ⟨5, 3⟩ = 5·⟨1, 0⟩ + 3·⟨0, 1⟩ = 5i + 3j
- ⟨7, −2⟩ = 7i − 2j
- 3·⟨1, 2⟩ + 2·⟨−1, 1⟩ = ⟨3, 6⟩ + ⟨−2, 2⟩ = ⟨1, 8⟩
8. Applications
Combining velocities and forces is a classic vector context:
📘 Example — Resultant velocity
A boat heads east at 8 m/s through water that flows north at 3 m/s.
- Boat's velocity vector: ⟨8, 0⟩
- Current's velocity vector: ⟨0, 3⟩
- Resultant: ⟨8, 3⟩
- Speed (magnitude): √(64 + 9) = √73 ≈ 8.54 m/s
- Direction angle: arctan(3/8) ≈ 20.6° north of east
9. Summary
- Add or subtract vectors by adding/subtracting components
- Geometric: head-to-tail rule or parallelogram rule
- Scalar multiplication scales magnitude and may reverse direction
- Vector operations satisfy the same algebraic properties as numbers (commutative, associative, distributive)
- Resultant vectors arise naturally in physics: velocities, forces, displacements