Study Guide: Vectors in Three Dimensions

Learning Objectives

After studying this chapter, you should be able to:

• Describe three-dimensional space mathematically using the rectangular coordinate system.
• Locate and plot points in space using $(x, y, z)$ coordinates.
• Calculate the distance between two points in $\mathbb{R}^3$ using the distance formula.
• Write and interpret equations for simple planes (parallel to coordinate planes) and spheres.
• Perform fundamental vector operations in three dimensions, including addition, scalar multiplication, and finding magnitudes.

Key Terms & Glossary

• $\mathbb{R}^3$ (Three-Space): The set of all ordered triples of real numbers, representing three-dimensional space.
• Origin: The point $(0, 0, 0)$ where the $x$, $y$, and $z$ axes intersect.
• Coordinate Planes: The planes formed by pairs of axes: the $xy$-plane ($z=0$), $xz$-plane ($y=0$), and $yz$-plane ($x=0$).
• Octant: One of the eight regions into which the three coordinate planes divide space.
• Component Form: Representing a vector as $\vec{v} = \langle v_1, v_2, v_3 \rangle$.
• Standard Unit Vectors: The vectors $\mathbf{i} = \langle 1, 0, 0 \rangle$, $\mathbf{j} = \langle 0, 1, 0 \rangle$, and $\mathbf{k} = \langle 0, 0, 1 \rangle$.
• Magnitude (Norm): The scalar length of a vector, denoted $|\vec{v}|$.

The "Big Idea"

In two dimensions, we can map out a flat surface like a piece of paper or a floor plan. However, the physical world involves depth, height, and breadth. By adding a third axis—the $z$-axis—perpendicular to the $xy$-plane, we transition from "flat" geometry to spatial geometry. This allows us to model complex systems such as planetary orbits, fluid flow, and structural engineering forces that do not lie in a single plane.

Formula / Concept Box

Hierarchical Outline

1. The Three-Dimensional Coordinate System
   • Axes and Origin: $x, y, z$ axes meet at $(0,0,0)$.
   • The Right-Hand Rule: Standard orientation where curling fingers from $x$ to $y$ leaves the thumb pointing toward $z$.
   • Planes and Octants: Space is divided into 8 octants by 3 planes ($xy, yz, xz$).
2. Points and Surfaces in Space
   • Distance Formula: An extension of the Pythagorean theorem to 3D.
   • Spheres: The 3D version of a circle; defined by all points equidistant from a center.
   • Simple Planes: Equations like $z = k$ represent a horizontal plane $k$ units from the $xy$-plane.
3. Vectors in $\mathbb{R}^3$
   • Component Form vs. Unit Vector Form: $\langle a, b, c \rangle$ vs $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$.
   • Algebraic Operations: Addition, subtraction, and scalar multiplication are performed component-wise.
   • Magnitude and Direction: Calculating how long a vector is and normalizing it to a unit vector.

Visual Anchors

3D Vector Concept Map

The 3D Rectangular Coordinate System

Definition-Example Pairs

• Term: Scalar Multiplication
  • Definition: Multiplying a vector by a real number $k$, which scales its magnitude and potentially reverses its direction.
  • Example: If $\vec{v} = \langle 2, -1, 4 \rangle$ and $k = 3$, then $3\vec{v} = \langle 6, -3, 12 \rangle$. The new vector is 3 times longer but points in the same direction.
• Term: Distance Formula in 3D
  • Definition: The straight-line distance between two points $P_1$ and $P_2$ in space.
  • Example: The distance between $(1, 0, 2)$ and $(4, 4, 2)$ is $\sqrt{(4-1)^2 + (4-0)^2 + (2-2)^2} = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{25} = 5$.
• Term: Sphere
  • Definition: The set of all points $(x, y, z)$ at a fixed distance $r$ from a center $(h, k, l)$.
  • Example: A soap bubble centered at $(0, 0, 0)$ with a radius of 2 inches is described by $x^2 + y^2 + z^2 = 4$.

Worked Examples

Example 1: Finding the Equation of a Sphere

Problem: Find the standard equation of a sphere with center $C(-2, 4, 7)$ and radius $r = 5$. Step 1: Identify the coordinates of the center: h = -2, k = 4, l = 7. Step 2: Identify the radius: $r = 5$. Step 3: Substitute into the standard sphere equation: $(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$. Step 4: Simplify the signs: $(x - (-2))^2 + (y - 4)^2 + (z - 7)^2 = 5^2$. Result: $(x + 2)^2 + (y - 4)^2 + (z - 7)^2 = 25$.

Example 2: Vector Magnitude and Normalization

Problem: Given $\vec{v} = \langle 3, -2, 6 \rangle$, find its magnitude and a unit vector in the same direction. Step 1 (Magnitude): $|\vec{v}| = \sqrt{3^2 + (-2)^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$. Step 2 (Unit Vector): $\vec{u} = \frac{\vec{v}}{|\vec{v}|} = \frac{1}{7} \langle 3, -2, 6 \rangle$. Result: $|\vec{v}| = 7$; $\vec{u} = \langle \frac{3}{7}, -\frac{2}{7}, \frac{6}{7} \rangle$.

Checkpoint Questions

1. What are the coordinates of the projection of the point $(3, -5, 8)$ onto the $xy$-plane?
2. If a sphere is described by $x^2 + y^2 + z^2 - 4x + 6y = 3$, what is its center and radius? (Hint: Complete the square).
3. Find the magnitude of the vector $\vec{w} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}$.
4. True or False: The equation $y = 4$ in $\mathbb{R}^3$ represents a line.

[!TIP] When visualizing 3D points, always start at the origin and move along the $x$-axis first, then the $y$-axis, then the $z$-axis. It is much easier to keep track of your position this way than trying to "jump" directly to the point.

[!IMPORTANT] The magnitude of a vector is always a non-negative scalar. If you get a negative value, re-check your squares and square root calculations!