Vectors in the Plane

Learning Objectives

After studying this chapter, you should be able to:

• Describe a plane vector using correct mathematical notation.
• Perform basic vector operations including scalar multiplication, addition, and subtraction.
• Express any vector in its component form $\langle x, y \rangle$.
• Calculate and explain the formula for the magnitude of a vector.
• Normalize a vector by expressing it in terms of standard unit vectors $\mathbf{i}$ and $\mathbf{j}$.
• Identify real-world examples of vector quantities like force and velocity.

Key Terms & Glossary

• Vector: A quantity possessing both magnitude (size) and direction.
• Scalar: A real number used to scale a vector; it has magnitude but no direction.
• Initial Point: The starting point of a directed line segment (the "tail").
• Terminal Point: The endpoint of a directed line segment (the "head").
• Magnitude: The length of the vector, denoted by $\|\mathbf{v}\|$.
• Zero Vector: A vector with a magnitude of zero and no specific direction, denoted $\mathbf{0}$.
• Equivalent Vectors: Vectors that have the same magnitude and direction, regardless of their position in the plane.

The "Big Idea"

Vectors are the fundamental language of multivariable calculus and physics. While a scalar tells us "how much" (like temperature or mass), a vector tells us "how much" and "which way" (like the thrust of a jet engine). By moving from simple numbers to vectors, we can model complex physical systems in two and three dimensions, allowing us to decompose motion into independent horizontal and vertical components.

Formula / Concept Box

Hierarchical Outline

1. Introduction to Vectors
   • Definition: Magnitude + Direction.
   • Notation: Boldface $\mathbf{v}$ or $\vec{v}$.
2. Vector Representation
   • Geometric: Directed line segments (arrows).
   • Algebraic: Component form $\langle x, y \rangle$.
   • Standard Basis: Linear combination $x\mathbf{i} + y\mathbf{j}$.
3. Basic Operations
   • Addition: Geometrically follows the Parallelogram Rule.
   • Subtraction: Adding the opposite vector $(\mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}))$.
   • Scalar Multiplication: Changes length; reverses direction if $k < 0$.
4. Magnitude and Unit Vectors
   • Distance Formula: Derived from the Pythagorean theorem.
   • Normalization: Dividing a vector by its own magnitude to get a unit length.

Visual Anchors

Vector Components and Geometry

Vector Addition (Triangle Method)

Definition-Example Pairs

• Equivalent Vectors: Vectors with identical magnitude and direction.
  • Example: An airplane flying at 500 mph North at 10,000 ft and another at 500 mph North at 20,000 ft have equivalent velocity vectors.
• Scalar Multiplication: Multiplying a vector by a real number.
  • Example: If $\mathbf{v}$ represents a force of 10N East, then $2\mathbf{v}$ represents a force of 20N East.
• Unit Vector: A vector with magnitude exactly 1.
  • Example: The vector $\langle 0.6, 0.8 \rangle$ is a unit vector because $\sqrt{0.6^2 + 0.8^2} = \sqrt{0.36 + 0.64} = 1$.

Worked Examples

Example 1: Finding Component Form and Magnitude

Problem: Find the component form and magnitude of vector $\mathbf{v}$ with initial point $P(-1, 3)$ and terminal point $Q(2, 7)$.

Solution:

1. Component Form: $\mathbf{v} = \langle x_Q - x_P, y_Q - y_P \rangle = \langle 2 - (-1), 7 - 3 \rangle = \langle 3, 4 \rangle$.
2. Magnitude: $\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

[!TIP] Always subtract (Terminal - Initial) to get the correct direction.

Example 2: Finding a Unit Vector

Problem: Find a unit vector $\mathbf{u}$ in the direction of $\mathbf{v} = \langle 5, -12 \rangle$.

Solution:

1. Calculate magnitude: $\|\mathbf{v}\| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
2. Divide by magnitude: $\mathbf{u} = \frac{1}{13}\langle 5, -12 \rangle = \langle \frac{5}{13}, -\frac{12}{13} \rangle$.

Checkpoint Questions

1. What is the difference between a scalar and a vector? Provide one example of each from physics.
2. If $\mathbf{u} = \langle 2, -3 \rangle$ and $\mathbf{v} = \langle 1, 5 \rangle$, find $2\mathbf{u} + \mathbf{v}$.
3. True or False: The zero vector $\mathbf{0}$ has a magnitude of 1 and points toward the origin.
4. How do you find the unit vector in the direction of any non-zero vector $\mathbf{w}$?

▶Click to see Answers

1. A scalar has only magnitude (e.g., speed), while a vector has magnitude and direction (e.g., velocity).
2. $2\langle 2, -3 \rangle + \langle 1, 5 \rangle = \langle 4, -6 \rangle + \langle 1, 5 \rangle = \langle 5, -1 \rangle$.
3. False. The zero vector has magnitude 0 and no defined direction.
4. Divide the vector $\mathbf{w}$ by its magnitude $\|\mathbf{w}\|$.