# PDF Text Book of 3-D Coordinate Systems and Straight Lines

The direction cosines are the cosines of the angles between a line and the coordinate axis. If we have a vector a, b, c in three dimensional space, then the direction cosines of the vector are defined as. While the direction cosines of a line segment are always unique, the direction ratios are never unique and in fact they can be infinite in number.

If the direction cosines of a line segment AB are l, m, n then those of line BA will be -l, -m, -n. Angle Between Two Lines. Also if the direction ratios of two lines a 1 , b 1 and c 1 and a 2 , b 2 and c 2 then the angle between two lines is given by. What is the projection of a line segment on a given line? A sphere is basically a circle in three dimensions. The general equation of sphere in 3D is. Hence, infinite number of triplets a 1 -a 1 -2a 1 are possible. Solution: Given equation of the straight line is. Hence, the point 4, 2, k must satisfy the plane which yields.

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## Line (geometry) - Wikipedia

Studying in Grade 6th to 12th? These points form a half-cone Figure. Rewrite the middle terms as a perfect square. This set of points forms a half plane.

## Cylindrical coordinate system

These points form a half-cone. Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. Express the location of Columbus in spherical coordinates. Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand.

A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations.

1. Writing Equations in \(ℝ^3\).
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6. History of geometry.

In the following example, we examine several different problems and discuss how to select the best coordinate system for each one. In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem.

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Which coordinate system is most appropriate for creating a star map, as viewed from Earth see the following figure? Cylindrical Coordinates When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension.

In three dimensions, this same equation describes a half-plane.

Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. Hint Converting the coordinates first may help to find the location of the point in space more easily.

Answer b This set of points forms a half plane. Find the center of gravity of a bowling ball.

Determine the velocity of a submarine subjected to an ocean current. Calculate the pressure in a conical water tank. Find the volume of oil flowing through a pipeline. Determine the amount of leather required to make a football. The origin should be located at the physical center of the ball. Specifies the type of axes, one of: boxed , frame , none , or normal. Font for the labels on the tick marks of the axes, specified in the same manner as font.

• Deixis and Alignment: Inverse Systems in Indigenous Languages of the Americas.
• Three-Dimensional Coordinate Systems;