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Cartesian coordinates in three dimensions
A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinatesX = 2, Y = 3, and Z = 4, or (2,3,4).
Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point p are obtained by drawing a line through p perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines.
Alternatively, the coordinates of a point p can also be taken as the (signed) distances from p to the three planes defined by the three axes. If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes. The distance is to be taken with the + or − sign, depending on which of the two half-spaces separated by that plane contains p. The y andz coordinates can be obtained in the same way from the (x,z) and (x,y) planes, respectively. The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x=1, the blue plane shows the points with z=1, and the yellow plane shows the points with y=−1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1).
DRAW COMMANDS WITH THERE CORRESPONDING OPTIONS
Toolbar
Draw
Pull-down
DrawLine
Keyboard
LINE
short-cut
L
With the Line command you can draw a simple line from one point to another. When you pickthe first point and move the cross-hairs to the location of the second point you will see a
A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinatesX = 2, Y = 3, and Z = 4, or (2,3,4).
Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point p are obtained by drawing a line through p perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines.
Alternatively, the coordinates of a point p can also be taken as the (signed) distances from p to the three planes defined by the three axes. If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes. The distance is to be taken with the + or − sign, depending on which of the two half-spaces separated by that plane contains p. The y andz coordinates can be obtained in the same way from the (x,z) and (x,y) planes, respectively. The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x=1, the blue plane shows the points with z=1, and the yellow plane shows the points with y=−1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1).
DRAW COMMANDS WITH THERE CORRESPONDING OPTIONS
Toolbar
Draw
Pull-down
DrawLine
Keyboard
LINE
short-cut
L
With the Line command you can draw a simple line from one point to another. When you pickthe first point and move the cross-hairs to the location of the second point you will see a