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{x = 0 } or coordinate planes {x = 0 },{y = 0 },{z = 0 }. In two dimensions, the x-coordinate usually directs to the ”east” and the y-coordinate points ”north”. In three dimensions, the usual coordinate system has the xy-plane as the ”ground” and the z-coordinate axes pointing ”up”.
Instead of using x,y- coordinates, we can also use the distance from the point to origin ⇢ and the angle it is away from the positive x-axis . Their relations are ⇢ = p x2+y2, tan = y/x, x2 R,y2 R; and conversely, x = ⇢cos ,y = ⇢sin ,⇢2 [0,1), 2 [0,2⇡). Lecture 3 stopped here. MATH 2D 9. Example 3.1.
Distance Formula We can derive a formula for the distance between two point in the plane using Pythagoras’ theorem. On can see form the following picture that the distance between the points A and B, denoted d(A;B) is given by d(A;B) = p (x 2 x 1)2 + (y 2 y 1)2; where A = (x 1;y 1) and B = (x 2;y 2). 18 16 14 12 10 8 6 4 2 Ð 2 Ð 4 Ð 6 Ð 8 ...
Finding Distance on the Coordinate Plane. You can think of a point as a dot, and a line as a series of points. In coordinate geometry you describe a point by an ordered pair (x, y), called the coordinates of the point. y-axis.
4) To get the distance between two planes ~n~x = d and ~n~x = e, then their distance is d( ;) = je dj j~nj Non-parallel planes have distance 0. Proof: use the distance formula between point and plane. For example, 5x+ 4y + 3z = 8 and 10x+ 8y + 6z = 2 have the distance j8 1j j[5;4;3]j = 7 p 50: Figure 1. The global positioning system GPS uses ...
CARTESIAN COORDINATE SYSTEMS. Points on the line are labeled by one coordinate x, a point in the plane is xed by 2 coordinates (x; y) and points in space are determined by three coordinates (x; y; z). 1D space = line =2 half lines. 2D space = plane =4 quadrants. 3D space = space =8 octants. y. z. (x,y) (x,y,z) y. x. CHOICE OF COORDINATE SYS-
The two most fundamental axioms of plane geometry are Axiom (1) Two (distinct nonparallel) lines intersect in a (unique) point. Axiom (2) Two (distinct) points determine a line.
