Three-Dimensional (3D) Space in Mathematics

 

Three-Dimensional (3D) Space in Mathematics

Three-dimensional (3D) space is a crucial mathematical concept, where objects or points are defined by three coordinates. It is commonly used in geometry, vector calculus, linear algebra, and calculus. Here's a detailed overview of 3D space in mathematics, covering key concepts, coordinate systems, geometric shapes, vector operations, and calculus.

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1. Definition of 3D Space

In mathematics, 3D space refers to a space that has three dimensions: length, width, and height (or depth). Any point in this space can be represented using three coordinates, $(x, y, z)$ which define its position in relation to three mutually perpendicular axes:

• x-axis (horizontal axis)
• y-axis (another horizontal axis, perpendicular to the x-axis)
• z-axis (vertical axis)

A point in 3D space is written as an ordered triplet $(x,y,z)<br><br>$

where:

• x-coordinate defines the position along the x-axis (left or right).
• y-coordinate defines the position along the y-axis (front or back).
• z-coordinate defines the position along the z-axis (up or down).

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2. Coordinate Systems in 3D

There are several ways to describe points and objects in 3D space. The most common coordinate systems are the Cartesian coordinate system and the cylindrical and spherical coordinate systems.

a) Cartesian Coordinate System

• In the Cartesian coordinate system, any point is represented by its position along the x, y, and z axes: $(x, y, z)$.
• It is an extension of the 2D Cartesian system with the addition of a third dimension (z-axis).
• Commonly used in geometry, algebra, and vector calculus.

b) Cylindrical Coordinate System

• A cylindrical coordinate system is useful for dealing with objects that have symmetry about an axis (e.g., cylinders).
• A point is described by the radial distance $r$, the azimuthal angle $\theta$ in the xy-plane, and the height $z$ (vertical distance): $$(r,\theta ,z)$$
• The relations between Cartesian and cylindrical coordinates are: $$x = r \cos \theta \quad , \quad y = r \sin \theta \quad , \quad z = z$$$$r = \sqrt{x^2 + y^2} \quad , \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$$

c) Spherical Coordinate System

• A spherical coordinate system is useful for dealing with problems involving spheres or spherical symmetry.
• Points are represented by a radial distance $r$, a polar angle $\theta$ (angle from the positive z-axis), and an azimuthal angle $\phi$ (angle in the xy-plane): $$(r, \theta, \phi)$$
  
• The relations between Cartesian and spherical coordinates are: $$x = r \sin \theta \cos \phi \quad , \quad y = r \sin \theta \sin \phi \quad , \quad z = r \cos \theta$$$$r = \sqrt{x^2 + y^2 + z^2} \quad , \quad \theta = \cos^{-1}\left(\frac{z}{r}\right) \quad , \quad \phi = \tan^{-1}\left(\frac{y}{x}\right)$$

3. Geometric Shapes in 3D

There are several basic shapes and objects in 3D space, each with its own properties related to volume, surface area, and symmetry.

a) Planes and Lines in 3D

• A line in 3D can be described parametrically using:

  $$\vec{r}(t) = \vec{r}_0 + t\vec{v}$$

  where $\vec{r}_0$ is a point on the line, $\vec{v}$ is a direction vector, and $t$ is a parameter.

• A plane in 3D can be described by the general equation:

  $$ax + by + cz = d$$

  where $a,b,\mathrm{c\; are\; the\; normal\; vector\; components\; to\; the\; plane,\; and }$$d$ is a constant.

b) Basic 3D Shapes

• Sphere: Set of points equidistant from a center point. The equation of a sphere with center $(h, k, l)$ and radius $r$ is: $$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$$
• Cube: A regular solid with six equal square faces. If the side length is $s$, the volume is $V = s^3$, and the surface area is $A = 6s^2$.
• Cylinder: A solid with circular cross-sections and parallel sides. The volume is: $$V = \pi r^2 h$$ where $r$ is the radius of the base, and $h$ is the height.
• Cone: A solid with a circular base and a single vertex. The volume is: $$V = \frac{1}{3} \pi r^2 h$$
• Pyramid: A polyhedron with a polygon base and triangular faces converging at a single vertex. The volume is: $$V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$$

c) Volume and Surface Area

• Volume of a Sphere: $$V = \frac{4}{3} \pi r^3$$
• Surface Area of a Sphere: $$A = 4 \pi r^2$$
• Volume of a Cylinder: $$V = \pi r^2 h$$
• Surface Area of a Cube: $$A = 6s^2$$

4. Transformations in 3D

In 3D space, transformations can be applied to objects, including translations, rotations, reflections, and scaling.

a) Translation

• Moves an object by a certain distance along the x, y, and z axes. $$(x', y', z') = (x + a, y + b, z + c)$$ where $a$, $b$, and $c$ are the translation distances.

b) Rotation

• Rotations in 3D can occur about the x, y, or z axes.
  • Rotation about the z-axis: $$x' = x \cos \theta - y \sin \theta \quad , \quad y' = x \sin \theta + y \cos \theta \quad , \quad z' = z$$
  • Rotation about the y-axis: $$z' = z \cos \theta - x \sin \theta \quad , \quad x' = z \sin \theta + x \cos \theta$$
  • Rotation about the x-axis: $$y' = y \cos \theta - z \sin \theta \quad , \quad z' = y \sin \theta + z \cos \theta$$

c) Scaling

• Scaling changes the size of an object by multiplying each coordinate by a scaling factor: $$(x', y', z') = (sx, sy, sz)$$

5. Vectors in 3D

A vector in 3D space is an entity that has both magnitude and direction. A vector is represented by its components along the x, y, and z axes:

$$\vec{v} = (v_x, v_y, v_z)$$

where $v_x$, $v_y$, and $v_z$are the components in the x, y, and z directions.

a) Vector Operations

• Vector Addition: $$\vec{v} + \vec{w} = (v_x + w_x, v_y + w_y, v_z + w_z)$$
• Scalar Multiplication: $$k\vec{v} = (k v_x, k v_y, k v_z)$$
• Dot Product: $$\vec{v} \cdot \vec{w} = v_x w_x + v_y w_y + v_z w_z$$
• Cross Product (produces a vector perpendicular to both vectors): $$\vec{v} \times \vec{w} = (v_y w_z - v_z w_y, v_z w_x - v_x w_z, v_x w_y - v_y w_x)$$

b) Magnitude of a Vector:

The magnitude (length) of a vector is:

$$|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$$

6. Calculus in 3D

In 3D, functions often depend on three variables, $f(x,y,z)\; representing\; surfaces\; or\; scalar\; fields\; in\; space.$

a) Partial Derivatives:

The partial derivative of a function $f(x, y, z)$with respect to $x$, $y$, or $z$measures the rate of change along one of the three directions.

b) Gradient:

The gradient of a scalar function $f(x, y, z)$ is a vector pointing in the direction of the steepest increase:

$$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$

c) Divergence and Curl:

• Divergence of a vector field $\vec{F} = (F_x, F_y, F_z)$ is a scalar representing the rate of expansion of the field: $$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$
• Curl of a vector field represents the rotation of the field: $$\nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)$$

Applications of 3D Space in Mathematics

• Geometry: Study of solid figures like spheres, cubes, and pyramids.
• Physics: Modeling forces, velocities, and other vector quantities in real-world space.
• Linear Algebra: Using matrices to perform transformations in 3D.
• Computer Graphics: Representing 3D objects on 2D screens using projection techniques.