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.
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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.
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This option overrides values specified for font. Specifies information about the x -axis and y -axis. The coordinate system used for display of the axes. By default, Cartesian axes are displayed. Note : This option is only available in the Standard interface. In the Classic interface, the coordinates are always Cartesian.
The background image or color for the plot. A plot can only have a single background image or color. The caption for the plot. The default is no caption. This option defines the font for a plot caption, specified in the same manner as font. Allows you to specify the color of the curves to be plotted. Allows you to apply a color scheme to a surface or set of points. When that is the case, then r The coordinate system. Allows detection of discontinuities. The options in the list are applied only to the filled area, and not to the original curve itself. This option does not work with non-Cartesian coordinate systems.
This option is valid only with the following commands: contourplot , implicitplot , and listcontplot. The final value, size , is the point size to be used. This option specifies labels for the axes. The default labels are the names of the variables in the original function to be plotted, if these are available; otherwise, no labels are used.source
12.7: Cylindrical and Spherical Coordinates
This option specifies the direction in which labels are printed along the axes. The default direction of any labels is horizontal. Font for the labels on the axes of the plot, specified in the same manner as font. Legend entry for a plot. Note that a set cannot be used as it does not preserve the order of the legend entries.
Legend style for a plot. Controls the line style of curves. Specifies the minimum number of points to be generated.