Question

In Exercises 3 and \(4,\) find the coordinates of the point. The point is located three units behind the \(y z\) -plane, four units to the right of the \(x z\) -plane, and five units above the \(x y\) -plane.

Short answer

The coordinates of the point are (-3, 4, 5).

Step-by-step solution

Identify the x-coordinate

Each unit behind the yz-plane represents one unit in the negative x direction. Given that the point is three units behind the yz-plane, we can conclude that the x-coordinate is -3.

Identify the y-coordinate

Each unit to the right of the xz-plane represents one unit in the positive y direction. Given that the point is four units to the right of the xz-plane, we can conclude that the y-coordinate is 4.

Identify the z-coordinate

Each unit above the xy-plane represents one unit in the positive z direction. Given that the point is five units above the xy-plane, we can conclude that the z-coordinate is 5.

Key concepts

Cartesian Coordinates

The Cartesian coordinate system is a foundational element in understanding geometry, especially within a three-dimensional space. This system allows us to pinpoint the exact location of a point by using three numbers, which are known as coordinates. Each number corresponds to a position along an axis: the x-axis, y-axis, and z-axis. These axes, which are perpendicular to each other, define the three dimensions of space.

In the exercise given, the task is to find the coordinates of a specific point in space. By knowing how each axis interacts with the physical space described—behind the yz-plane, right of the xz-plane, and above the xy-plane—we can determine these coordinates. To simplify, we relate the directions (behind, to the right, and above) to negative or positive movements along the respective axis. Remember, moving 'behind' the yz-plane means moving in the negative x direction, 'to the right' of the xz-plane implies a positive y direction, and 'above' the xy-plane suggests a positive z direction. This intuitive mapping simplifies complex spatial problems into a set of numbers that are easy to work with.

Spatial Reasoning

Spatial reasoning involves the ability to understand and remember the spatial relations among objects. This skill is crucial when translating verbal descriptions of physical space into a geometric representation, such as when finding coordinates on a 3D grid. For instance, when we say a point is 'three units behind the yz-plane,' we must picture which way 'behind' is relative to the yz-plane within a three-dimensional context.

To enhance spatial reasoning, it's helpful to visualize the axes of the three-dimensional coordinate system extending in all directions. Imagining the axes as guidelines, we can mentally 'move' along these paths to reach the desired point in space. In doing so, we improve our ability to navigate and solve problems within three-dimensional spaces—both in academic settings and real-world applications like architecture, engineering, and design.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, bridges algebra and geometry by using coordinates to describe geometric shapes. It offers a powerful toolset for solving geometric problems by expressing them in algebraic terms. Instead of relying solely on visual proofs, analytic geometry provides a numerical approach that can be more precise and versatile.

The exercise we're looking at is a practical application of analytic geometry. By associating coordinates with physical locations, we anchor abstract numbers to the real world. This application extends beyond simple points, as we can also describe lines, planes, and various shapes within the Cartesian coordinate system, setting the foundation for further exploration in fields like calculus, physics, and computer graphics where such geometric understanding is essential.