Lesson 1
Introduction to Three
Dimensional (3D)
Geometry; Plane
Sketching
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Introduction to Three-Dimensional Geometry: Coordinate Systems and Plane Sketching, Slides of Calculus for Engineers
Three-dimensional (3d) geometry, focusing on plotting points, determining distances, and sketching planes in a 3d coordinate system. It begins with a review of the two-dimensional coordinate system before transitioning to 3d space, explaining the x, y, and z axes and their corresponding planes. Examples of plotting points in 3d space and introduces equations for planes, including those parallel to the coordinate planes. It also covers distance calculations and midpoint formulas in 3d, providing a foundational understanding of spatial geometry. This material is suitable for students learning the basics of 3d geometry and spatial reasoning, providing a clear and structured introduction to the topic. Designed to help students visualize and understand spatial relationships, which are crucial in various fields such as engineering, physics, and computer graphics.
Typology: Slides
2024/2025
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Lesson 1
Introduction to Three
Dimensional (3D)
Geometry; Plane
Sketching
Objectives:
- At the end of the lesson, the student must be able
to :
- Plot points in three dimensional system.
- Determine the distance between two points.
- Determine the point of division and the
midpoint of a line segment.
- Sketch a plane in three dimensional system.
Three Dimensional
Geometry
Let OX, OY, and OZ be three mutually perpendicular lines. These lines constitute the x-axis, the y-axis, and the z-axis of a three- dimensional rectangular coordinate system. The axes, in pairs, determine three mutually perpendicular planes called coordinate planes. The planes are designated as the XOY- plane, the XOZ-plane, and the YOZ-plane or, more simply, the xy- plane, the xz-plane, and the yz-plane. The coordinate planes divide space into eight regions called octants. The distance of P from the yz-plane is called the x-coordinate , the distance from the xz-plane the y-coordinate , and the distance from the xy- plane the z-coordinate. The coordinates of a point are written in the form (x, y, z), in this order, x first, y second, and z third.
Coordinate Planes
The three coordinate axes
determine
the three coordinate planes.
The xy -plane contains the x - and y -axes. The yz -plane contains the y - and z -axes. The xz -plane contains the x - and z -axes.
Example
Plot the given points in a three-dimensional coordinate
system.
3D-Space Point-Plotting
Examples
Examples:
SURFACES : A. Plane
An equation of the form
Ax + By + Cz + D = 0
represents a plane.
a) x = k, plane parallel to yz-plane b) y = k, plane parallel to xz-plane c) z = k, plane parallel to xy-plane
d) Ax + By + D = 0, plane parallel to z- axis e) By + Cz + D = 0, plane parallel to x- axis
f) Ax + Cz + D = 0, plane parallel to y- axis