Welcome to Chapter 12: Vectors and the Geometry of Space!

This page contains all the resources you need to succeed in Chapter 12. Here, you'll find class notes, PowerPoint presentations, and video lectures covering topics from three-dimensional coordinate systems to vector operations. Let's dive in and explore the beauty of spatial mathematics!

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Class Notes

Section 12-1: Three-Dimensional Coordinate Systems

Let's start with the basics. In this section, we extend our familiar two-dimensional coordinate system to three dimensions. A point in 3D space is represented by an ordered triple $P(x, y, z)$. We'll cover:

  • The three coordinate planes: $xy$-plane, $yz$-plane, and $xz$-plane.
  • The distance formula in three dimensions: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$.
  • Equations of simple surfaces like planes (e.g., $z = 3$ is a plane parallel to the $xy$-plane)

Section 12-2: Vectors

Vectors are crucial for representing quantities with both magnitude and direction. Here, we will discuss:

  • Basic vector operations: addition, subtraction, and scalar multiplication. If $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$, then $\mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle$.
  • The concept of unit vectors, especially the standard basis vectors $\mathbf{i} = \langle 1, 0, 0 \rangle$, $\mathbf{j} = \langle 0, 1, 0 \rangle$, and $\mathbf{k} = \langle 0, 0, 1 \rangle$. Any vector can be expressed as a linear combination of these: $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$.
  • Finding the vector from point $A(x_1, y_1, z_1)$ to point $B(x_2, y_2, z_2)$: $\overrightarrow{AB} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$.

Section 12-3: The Dot Product

The dot product provides a way to multiply vectors and obtain a scalar. Key ideas include:

  • Definition: If $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$, then $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$.
  • The geometric interpretation: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos{\theta}$, where $\theta$ is the angle between the vectors.
  • Using the dot product to determine if vectors are orthogonal (perpendicular): $\mathbf{a} \cdot \mathbf{b} = 0$ if and only if $\mathbf{a}$ and $\mathbf{b}$ are orthogonal.

Section 12-4: The Cross Product

The cross product yields a vector that is orthogonal to both input vectors. Learn about:

  • Definition: If $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$, then $\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$. This can be easier to compute using determinants.
  • The cross product is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$.
  • The magnitude of the cross product: $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin{\theta}$, which is equal to the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$.

Keep practicing, and don't hesitate to reach out with any questions. Let's conquer Chapter 12 together!