Glossary

1.1 Three-Dimensional Coordinate Systems

2 min read•july 25, 2024

expand our understanding of space beyond flat surfaces. They use three perpendicular to pinpoint locations, creating a framework for describing position, movement, and relationships in 3D.

Points, planes, and distances in 3D space build on familiar 2D concepts. The extends to include a , while midpoints are found by averaging x, y, and z values. These tools help us navigate and analyze complex spatial relationships.

Three-Dimensional Coordinate Systems

Components of 3D coordinate systems

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  • Three mutually perpendicular axes intersect at point (0, 0, 0) form foundation of 3D space
    • extends horizontally representing left-right movement
    • extends vertically representing up-down movement
    • extends into/out of plane representing depth/height
  • divide 3D space into eight regions created by axis intersections (+x+y+z, +x+y-z, etc.)
  • form flat surfaces where one coordinate equals zero
    • (z = 0) represents horizontal surface
    • (y = 0) represents front-to-back vertical surface
    • (x = 0) represents side-to-side vertical surface
  • determines positive axis directions (thumb-x, index-y, middle-z)

Points in 3D space

  • (x, y, z) uniquely identifies point locations
  • measures distance from yz-plane (left/right)
  • measures distance from xz-plane (up/down)
  • z-coordinate measures distance from xy-plane (forward/backward)
  • Positive and negative values indicate direction from origin
  • Special cases illustrate point positions
    • Points on coordinate planes have one zero coordinate (3, 0, 5)
    • Points on coordinate axes have two zero coordinates (0, 4, 0)

Distance between 3D points

  • Distance formula in 3D extends 2D version: d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
  • Pythagorean theorem generalized to three dimensions
  • Calculate distance:
    1. Identify coordinates of two points (x₁, y₁, z₁) and (x₂, y₂, z₂)
    2. Subtract corresponding coordinates
    3. Square the differences
    4. Sum the squared differences
    5. Take the square root of the sum

Midpoints of 3D line segments

  • in 3D: (x1+x22,y1+y22,z1+z22)(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})
  • Find midpoint:
    1. Identify coordinates of endpoints (x₁, y₁, z₁) and (x₂, y₂, z₂)
    2. Add corresponding coordinates
    3. Divide each sum by 2
  • Midpoint divides line segment into two equal parts, relates to distance formula

Key Terms to Review (19)

Axes: Axes are the reference lines used in a coordinate system to define the orientation of space and the positioning of points within that space. In a three-dimensional coordinate system, there are typically three axes: the x-axis, y-axis, and z-axis, which are mutually perpendicular and provide a framework for locating any point in three-dimensional space by using ordered triples.
Coordinate planes: Coordinate planes are two-dimensional surfaces defined by a pair of perpendicular axes, typically labeled as the x-axis and y-axis, which are used to represent points in a Cartesian coordinate system. In three-dimensional space, these planes become essential for visualizing the relationship between points, lines, and shapes as they provide a foundation for understanding how to navigate and interpret spatial information.
Distance Formula: The distance formula calculates the straight-line distance between two points in a three-dimensional space using their coordinates. It is derived from the Pythagorean theorem and extends its application from two dimensions into three, enabling us to find distances in a more complex setting. This formula is essential for understanding geometric relationships in 3D coordinate systems, as it provides a quantitative measure of how far apart points are in that space.
Midpoint formula: The midpoint formula is a mathematical tool used to find the exact middle point between two coordinates in a space. This formula is significant in three-dimensional coordinate systems as it allows us to determine the center of a line segment defined by its endpoints, enabling us to analyze geometric shapes, distances, and spatial relationships more effectively.
Octants: Octants are the eight distinct regions created by dividing three-dimensional space using the coordinate axes. Each octant represents a unique combination of positive and negative values for the x, y, and z coordinates, allowing for a structured way to describe the position of points in 3D space. Understanding octants is crucial for visualizing geometric concepts and for navigating through various applications of multivariable calculus.
Ordered triple notation: Ordered triple notation refers to a way of representing points in three-dimensional space using three coordinates, typically written in the form (x, y, z). This notation provides a clear way to identify the position of a point relative to the three axes: the x-axis, y-axis, and z-axis. Each coordinate corresponds to a specific distance from the origin along its respective axis, enabling precise location in a three-dimensional coordinate system.
Origin: The origin is the point in a three-dimensional coordinate system where the x, y, and z coordinates are all equal to zero, represented as (0, 0, 0). It serves as the reference point for locating all other points in space and is crucial for understanding how to plot and interpret points in three dimensions. The origin also helps establish the direction of the axes and can be seen as the intersection of these axes.
Pythagorean Theorem in 3D: The Pythagorean Theorem in 3D extends the concept of the classic theorem, which relates the sides of a right triangle, to three-dimensional space. It states that for any right rectangular prism (or rectangular box), the square of the length of the diagonal connecting opposite corners is equal to the sum of the squares of the lengths of the three edges that meet at one corner. This theorem helps in calculating distances between points in a three-dimensional coordinate system.
Right-Hand Rule: The right-hand rule is a mnemonic used to determine the direction of a vector in three-dimensional space, particularly when dealing with cross products. It helps to establish a consistent orientation for vector quantities, making it easier to visualize and solve problems in three-dimensional coordinate systems. By using the right hand to align fingers with two vectors, the thumb points in the direction of the resultant vector, establishing a clear spatial relationship among them.
Three-dimensional coordinate systems: A three-dimensional coordinate system is a method for defining the position of points in space using three coordinates, typically represented as (x, y, z). This system allows for the visualization and analysis of geometric shapes and objects in three dimensions, which is essential for fields such as engineering, physics, and computer graphics. The coordinates correspond to distances along the three mutually perpendicular axes: the x-axis, y-axis, and z-axis.
X-axis: The x-axis is a horizontal line in a Cartesian coordinate system that represents the independent variable in a function. It is one of the primary axes used to define the position of points in a two-dimensional or three-dimensional space, intersecting with the y-axis and z-axis at the origin (0,0,0). The x-axis allows for the visualization of relationships and changes in values across various dimensions.
X-coordinate: The x-coordinate is a numerical value that represents the horizontal position of a point in a three-dimensional space. It is one of the three coordinates used to define a point's location, alongside the y-coordinate and z-coordinate. The x-coordinate indicates how far left or right a point is from the origin, which serves as the reference point in a Cartesian coordinate system.
Xy-plane: The xy-plane is a two-dimensional flat surface defined by the x-axis and y-axis, where any point can be represented by an ordered pair $(x, y)$. This plane is fundamental in mathematics and serves as the basis for understanding three-dimensional coordinate systems, where the addition of the z-axis allows for a more complete representation of spatial relationships.
Xz-plane: The xz-plane is a two-dimensional plane in three-dimensional space defined by the x and z coordinates, where the y-coordinate is always zero. This plane consists of all points that can be described by pairs of values (x, 0, z), which means any point on this plane will have no vertical displacement. Understanding the xz-plane is crucial for visualizing three-dimensional geometric relationships and analyzing surfaces and curves in multivariable contexts.
Y-axis: The y-axis is the vertical line on a two-dimensional or three-dimensional coordinate system, which helps to define the position of points in space. It plays a crucial role in representing the second variable in ordered pairs, while working in conjunction with the x-axis and z-axis to establish the framework for plotting and analyzing relationships among variables.
Y-coordinate: The y-coordinate is the second number in an ordered pair used to specify the position of a point in a two-dimensional or three-dimensional coordinate system. It represents the vertical position of a point relative to a defined origin, impacting how we visualize and interpret spatial relationships in different dimensions.
Yz-plane: The yz-plane is a two-dimensional plane in three-dimensional space where the x-coordinate is always equal to zero. This means that any point located on the yz-plane has its coordinates expressed as (0, y, z), indicating that it lies vertically along the y and z axes. The yz-plane is essential for visualizing three-dimensional objects and understanding spatial relationships, as it helps to represent how different dimensions interact with one another.
Z-axis: The z-axis is a crucial element in a three-dimensional coordinate system, representing the vertical dimension. It is perpendicular to both the x-axis and y-axis, allowing for the depiction of points in space with three coordinates: (x, y, z). This axis is essential for understanding spatial relationships and visualizing objects in 3D space.
Z-coordinate: The z-coordinate is a key component of the three-dimensional Cartesian coordinate system, representing the vertical position of a point in space. It works alongside the x-coordinate and y-coordinate to specify the location of points in 3D space, enabling a more comprehensive understanding of geometric relationships. The z-coordinate is crucial for visualizing and analyzing three-dimensional objects and phenomena.
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