Lesson 21 - 3D Geometry Basics, Slides of Calculus for Engineers

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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

Illustrative Problems

Illustrative Problems

Illustrative Problem