Learn & Review: Calculus 3 Lecture 11.1: Vectors in 2D

Introduction to Vectors and Their Operations

This summary outlines the fundamental concepts of vectors, their properties, and basic operations, as introduced in the provided lecture content. Vectors are presented as essential tools for describing motion and quantities that have both magnitude and direction, building upon the foundations of Calculus I and II.

Main Idea: The Importance and Application of Vectors

• Building on Calculus: Calculus III, while not fully utilizing all of Calculus I and II, provides crucial insights into their applications. Vectors are introduced as a foundational element for understanding more complex concepts in Calculus III.
• Describing Motion and Direction: Vectors are essential for describing motion in three dimensions and for providing more precise information than just speed or direction alone. They combine both aspects to accurately represent movement and forces.

Core Concepts of Vectors

• Definition: A vector is a mathematical object that describes both a magnitude (often referred to as speed or length) and a direction.
• Magnitude vs. Speed: While "speed" is an intuitive term, "magnitude" is the more precise mathematical term used for the length or size of a vector.
• Direction: Direction is typically represented on a coordinate system (like the x-y plane or x-y-z space).
• Representation: Vectors are often represented by arrows. In this context, they are denoted by a letter with a half-arrow on top (e.g., $\vec{v}$).
• Initial and Terminal Points: A vector has a starting point (initial point) and an ending point (terminal point). The order matters: a vector from A to B is different from a vector from B to A.

Operations with Vectors

1. Scalar Multiplication

• Scalar: A scalar is simply a number (a constant) that does not have a direction.
• Effect: Multiplying a vector by a scalar can:
  • Alter the length (magnitude): A scalar greater than 1 increases the magnitude, while a scalar between 0 and 1 decreases it.
  • Reverse the direction: Multiplying by a negative scalar reverses the vector's direction.
• Parallelism: Scalar multiples of a vector are always parallel to the original vector. This is because they maintain the same or opposite direction.

2. Vector Addition

• Graphical Method: Vectors are added by placing the initial point of the second vector at the terminal point of the first vector. The resultant vector goes from the initial point of the first vector to the terminal point of the second vector (end-to-end addition).
• Commutativity: Vector addition is commutative, meaning the order of addition does not affect the result ($\vec{v} + \vec{w} = \vec{w} + \vec{v}$). This is illustrated by the parallelogram law.
• Mathematical Method: Vector addition is performed component-wise. If $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$, then $\vec{a} + \vec{b} = \langle a_1 + b_1, a_2 + b_2 \rangle$.

3. Vector Subtraction

• Concept: Subtracting a vector is equivalent to adding its opposite (negative). $\vec{v} - \vec{w} = \vec{v} + (-\vec{w})$.
• Mathematical Method: Vector subtraction is also performed component-wise. If $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$, then $\vec{a} - \vec{b} = \langle a_1 - b_1, a_2 - b_2 \rangle$.

Position Vectors and Notation

• Position Vector: A special type of vector whose initial point is at the origin (0,0). Its terminal point defines a specific point in space.
• Notation:
  • Points: Represented by parentheses, e.g., (x, y).
  • Position Vectors: Represented by brackets, e.g., $\langle x, y \rangle$. This notation signifies a vector with magnitude and direction, originating from the origin.
• One-to-One Relationship: There is a one-to-one correspondence between points in space and their corresponding position vectors.
• Translating Vectors: Any vector can be translated to become a position vector. To find the position vector between two points (P1 to P2), subtract the coordinates of the initial point (P1) from the coordinates of the terminal point (P2): $\vec{v} = \langle x_2 - x_1, y_2 - y_1 \rangle$.

Properties of Position Vectors

• Slope: For a position vector $\langle x, y \rangle$, the slope is simply $\frac{y}{x}$. This is because the initial point is the origin.
• Magnitude: The magnitude of a position vector $\langle x, y \rangle$ is found using the distance formula from the origin: $||\vec{v}|| = \sqrt{x^2 + y^2}$. This is equivalent to the Pythagorean theorem.
• Equality: Two vectors are equal if and only if they have the same magnitude and the same slope (direction). Equivalently, their components must be identical.

Standard Basis Vectors (i and j)

• Definition:
  • $\vec{i} = \langle 1, 0 \rangle$: A unit vector pointing in the positive x-direction.
  • $\vec{j} = \langle 0, 1 \rangle$: A unit vector pointing in the positive y-direction.
• Representation: Any position vector $\langle x, y \rangle$ can be expressed using standard basis vectors as $x\vec{i} + y\vec{j}$. This represents moving $x$ units in the x-direction and $y$ units in the y-direction.
• Parallelism: Vectors are parallel if they are scalar multiples of each other. This means they have the same direction (or opposite directions) and can be scaled to match. This is a more general test for parallelism than just comparing slopes, especially in 3D.

Unit Vectors

• Definition: A unit vector is a vector with a magnitude of one.
• Purpose: Unit vectors primarily indicate direction.
• Finding a Unit Vector: To find the unit vector ($\hat{u}$) in the direction of a given vector ($\vec{v}$), divide the vector by its magnitude: $\hat{u} = \frac{\vec{v}}{||\vec{v}||}$.
• Vector Representation: Any vector $\vec{v}$ can be represented as the product of its magnitude and a unit vector in its direction: $\vec{v} = ||\vec{v}|| \cdot \hat{u}$.
• Unit Circle Connection: Unit vectors originating from the origin have their terminal points on the unit circle. If a unit vector makes an angle $\theta$ with the positive x-axis, its components are $(\cos \theta, \sin \theta)$, so the vector can be written as $(\cos \theta)\vec{i} + (\sin \theta)\vec{j}$.

Applications and Examples

• Physics Problems: Vectors are used to model forces, velocities, and displacements. For instance, analyzing forces on a hanging weight involves setting up equations based on vector addition and equilibrium.
• Navigation: Describing the plane's velocity and the wind's velocity requires vectors. The resultant vector indicates the plane's actual ground track and speed.

This summary covers the foundational concepts of vectors, including their definition, representation, scalar multiplication, addition, subtraction, position vectors, standard basis vectors, unit vectors, and parallelism, along with introductory applications.