Vectors and Spheres in Calculus: Coordinate Geometry and Applications

Vectors in 2D and 3D Space

Introduction to Vectors

Vectors are mathematical objects that have both magnitude and direction. They are fundamental in calculus, physics, and engineering for representing quantities such as displacement, velocity, and force.

• Vector Notation: A vector in 2D is written as $\vec{v} = \langle v_x, v_y \rangle$, and in 3D as $\vec{v} = \langle v_x, v_y, v_z \rangle$.

• Example: $\vec{v} = \langle -4, -3 \rangle$

Vector Magnitude and Normalization

The magnitude (or length) of a vector is a measure of its size. Normalization is the process of converting a vector to a unit vector (magnitude 1) in the same direction.

• Magnitude Formula (2D): $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$

• Magnitude Formula (3D): $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

• Example: For $\vec{v} = \langle -4, -3 \rangle$, $|\vec{v}| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5$

• Normalization: $\vec{u} = \frac{\vec{v}}{|\vec{v}|}$

• Example: $\vec{u} = \frac{1}{5} \langle -4, -3 \rangle = \langle -\frac{4}{5}, -\frac{3}{5} \rangle$

Coordinate Systems and the Right-Hand Rule

2D and 3D Coordinate Axes

Vectors are represented in coordinate systems. The 2D system uses x and y axes, while the 3D system adds the z axis.

• 2D Axes: x and y

• 3D Axes: x, y, and z

• Right-Handed Coordinate System: The orientation of axes follows the right-hand rule, where curling the fingers from x to y, the thumb points in the direction of z.

Vector Operations

Addition and Scalar Multiplication

Vectors can be added and multiplied by scalars (real numbers).

• Addition: $\langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle = \langle x_1 + x_2, y_1 + y_2 \rangle$

• Scalar Multiplication: $a \langle x, y \rangle = \langle a x, a y \rangle$

• 3D Addition: $\langle x_1, y_1, z_1 \rangle + \langle x_2, y_2, z_2 \rangle = \langle x_1 + x_2, y_1 + y_2, z_1 + z_2 \rangle$

• 3D Scalar Multiplication: $a \langle x, y, z \rangle = \langle a x, a y, a z \rangle$

Distance Between Points

The distance between two points in space is given by the magnitude of the vector connecting them.

• Distance Formula (3D): $|\overrightarrow{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

• Example: For $P = (0, -4, 3)$ and $Q = (-4, -2, 7)$, $|\overrightarrow{PQ}| = \sqrt{(-4 - 0)^2 + (-2 - (-4))^2 + (7 - 3)^2} = \sqrt{16 + 4 + 16} = 6$

Unit Vectors in Coordinate Directions

Unit vectors are vectors of length 1 pointing along the coordinate axes.

• Standard Unit Vectors:

  • $\vec{i}$: along x-axis

  • $\vec{j}$: along y-axis

  • $\vec{k}$: along z-axis

Equations of Circles and Spheres

Circle in 2D

A circle is the set of all points in a plane at a fixed distance (radius) from a center point.

• Equation: $(x - a)^2 + (y - b)^2 = r^2$

• Center: $C = (a, b)$

• Radius: $r$

Sphere in 3D

A sphere is the set of all points in space at a fixed distance (radius) from a center point.

• Equation: $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$

• Center: $C = (a, b, c)$

• Radius: $r$

Completing the Square for Spheres

Finding the Center and Radius

To find the center and radius of a sphere from a general quadratic equation, complete the square for each variable.

• Example: $x^2 + 8x + y^2 - 6y + z^2 - 11 = 0$

• Complete the squares:

  • $(x^2 + 8x + 16) + (y^2 - 6y + 9) + (z^2 - 0z + 0) = 0 + 16 + 9 + 11$

  • $(x + 4)^2 + (y - 3)^2 + (z - 0)^2 = 36$

• Center: $C = (-4, 3, 0)$

• Radius: $r = 6$

Antipodal Points and Spheres

Definition and Properties

Antipodal points on a sphere are pairs of points that are diametrically opposite each other.

• Diameter: The vector between antipodal points is a diameter of the sphere.

• Radius: Half the length of the diameter.

• Example: For $P = (0, -4, 3)$ and $Q = (-4, -2, 7)$:

  • $\overrightarrow{PQ} = \langle -4, 2, 4 \rangle$

  • $|\overrightarrow{PQ}| = 6$

  • Radius $r = \frac{1}{2} |\overrightarrow{PQ}| = 3$

Equation of Sphere from Antipodal Points

Given two antipodal points, the center is the midpoint, and the radius is half the distance between them.

• Center: $C = \frac{1}{2}(P + Q)$

• Radius: $r = \frac{1}{2}|\overrightarrow{PQ}|$

• Equation: $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$

Region Inside a Sphere

Definition

The region inside a sphere consists of all points whose distance from the center is less than or equal to the radius.

• Inequality: $(x - a)^2 + (y - b)^2 + (z - c)^2 \leq r^2$

• Example: For center $(-2, -3, 5)$ and radius $3$, the region is $(x + 2)^2 + (y + 3)^2 + (z - 5)^2 \leq 9$

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
Vector Magnitude (2D)	$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$	Length of vector in 2D
Vector Magnitude (3D)	$|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$	Length of vector in 3D
Normalization	$\vec{u} = \frac{\vec{v}}{|\vec{v}|}$	Unit vector in direction of $\vec{v}$
Distance Between Points (3D)	$|\overrightarrow{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$	Distance from $P$ to $Q$
Equation of Circle	$(x - a)^2 + (y - b)^2 = r^2$	All points at distance $r$ from $(a, b)$
Equation of Sphere	$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$	All points at distance $r$ from $(a, b, c)$
Region Inside Sphere	$(x - a)^2 + (y - b)^2 + (z - c)^2 \leq r^2$	All points within or on the sphere

Additional info: The notes above expand on the original handwritten content, providing full definitions, formulas, and context for vectors, coordinate systems, and spheres as encountered in introductory calculus and analytic geometry.