Calculus 3: Three-Dimensional Coordinate Systems (Video #1) | Math with Professor V

This paragraph introduces the first video in a series on Calculus 3, focusing on vectors and spatial geometry. It explains the transition from two-dimensional Cartesian coordinate systems, which require an ordered pair (X, Y), to three-dimensional systems requiring an ordered triple (X, Y, Z). The video script details the setup of a 3D coordinate system with perpendicular axes intersecting at the origin and describes the right-hand rule for orientation. It also explains the concept of coordinate planes (XY, XZ, YZ) and their respective equations, as well as the division of three-dimensional space into octants, with a focus on the first octant where all coordinates are positive. The paragraph concludes with a discussion on plotting points in 3D, emphasizing the importance of proper labeling and scaling of axes.

The second paragraph delves into the practical aspects of plotting points in three-dimensional space, using the points (3, 2, 5) and (4, -2, -1) as examples. It describes the process of moving along the x, y, and z axes to reach the desired coordinates, highlighting the importance of perspective and accurate scaling. The paragraph then moves on to graphing in 3D, contrasting it with 2D graphing, starting with the equation x = 2, which graphs as a vertical line in 2D and as a plane in 3D. It continues with the equation y = 3, which graphs as a horizontal line in 2D and also represents a plane in 3D, and y = x, which is a line with a 45-degree angle in 2D and extends as a plane in 3D. The final example involves the equation xy = 0, which corresponds to graphing both the YZ and XZ planes in 3D.

This paragraph explores more complex graphing scenarios in three-dimensional space. It starts with the graph of y = x^2, a parabola in 2D, and extends this concept to 3D by projecting the graph parallel to the z-axis, creating a 3D representation. The next example is x^2 + z^2 = 16, which is a circle in the XZ plane when y is omitted. The graph is projected parallel to the y-axis to form a 3D shape. The paragraph also discusses graphing three-dimensional inequalities, such as x ≤ y, which in 2D is represented by a region above the line y = x, and in 3D, it is represented by all points behind the plane y = x. The summary emphasizes the importance of understanding the relationship between 2D and 3D graphs and the techniques for extending them into three-dimensional space.

The fourth paragraph discusses the derivation of the three-dimensional distance formula by extending the two-dimensional version. It uses a rectangular solid with vertices at distinct points (x1, y1, z1) and (x2, y2, z2) to illustrate the process. By considering the differences in coordinates as sides of a right triangle and applying the Pythagorean theorem, the paragraph shows how to arrive at the 3D distance formula, which includes the square of the difference in z-coordinates. The summary also explains how the 2D distance formula leads to the equation of a circle, and similarly, the 3D formula leads to the equation of a sphere. Examples of writing the equation of a sphere with a given center and radius are provided, demonstrating the practical application of the formula.

In this paragraph, the process of completing the square for the equation of a sphere is explained, with an example that starts with an equation involving x, y, and z terms. By dividing through by 2 and adding the necessary constants to complete the square for each variable, the equation is transformed into the standard form of a sphere, revealing the center and radius. The summary also includes a description of a region defined by an inequality involving the sum of squared coordinates, which represents all points between two concentric spheres with radii 1 and 5, respectively. The paragraph concludes by emphasizing the ability to describe such regions in words without the need for a graphical representation.

The final paragraph wraps up the first video in the Calculus 3 series by summarizing the topics covered, including the 3D coordinate system, graphing in three dimensions, the derivation and application of the 3D distance formula, and the completion of the square for sphere equations. It emphasizes the importance of understanding these foundational concepts for further study in calculus and spatial geometry. The paragraph leaves the viewer with a clear understanding of the material presented and sets the stage for future videos in the series.