Download Lecture 1. Three-Dimensional Coordinate System. June 18

Three-Dimensional Coordinate System
Lecture 1. 06/18/2012
On the plane, we can represent any point by a pair of two Cartesian coordinates (x, y), which are
defined by two perpendicular axes: the x-axis and the y-axis. In the space, we have three Cartesian
coordinates (x, y, z) and three mutually perpendicular axes: x, y and z-axis. The direction of the
z-axis is determined by the right-hand rule: if you curl the fingers of your right hand from the
x-axis to the y-axis, provided the hand is centered at the origin (the point of their intersection),
then the thumb points to the direction of the z-axis.
There are three ways to interpret three-dimensional Cartesian coordinates.
1. Suppose P is any point in the space. Put the perpendicular P Q down to the x-axis. Then
the point Q lies on the x-axis.
- If it lies on the positive side, then the distance between O (the origin) and Q is called the
x-coordinate of P .
- If Q lies on the negative side, then minus this distance is called the x-coordinate of P .
- If Q coincides with O, then this coordinate is equal to zero and the point P lies on the
yz-coordinate plane.
2. Also, we can view the x-coordinate of P as the signed distance from P to this yz-coordinate
plane. The y- and z-coordinates are defined similarly.
3. Also, suppose we have a triple (x0 , y0 , z0 ). Start from the origin O. Go along the x-axis by
the distance x0 . This means:
- if x0 > 0 go in the direction of the x-axis and cover the distance x0 ;
- if x0 < 0 go in the opposite direction and cover the distance |x0 |;
- if x0 = 0 just stay still.
Reach some point Q, then go along the y-axis by the distance y0 , reach R, then go along the
z-axis by the distance z0 . You will reach the point P with coordinates (x0 , y0 , z0 ).
The set of all triples (x, y, z) of real numbers, or, in other words, the set of all points in the
space will be denoted by R3 . Just like the set of all pairs (x, y) of real numbers, i.e. the set of all
points on the plane will be denoted by R2 .
Example. In R3 , z = 0 is the xy-coordinate plane, y = 5 is the plane which is parallel to the
xz-coordinate plane and 5 units away from it, in the direction of the y-axis. y = 5, z = 3 is a line
parallel to the x-axis and 5 units away from the xz-plane and 3 units away from the xy-plane.
The distance formula. In R2 , the distance between the two points (x1 , y1 ) and (x2 , y2 )
p
is (x2 − x1 )2 + (y1 − y2 )2 . Similarly, in R3 , the distance between P1 = (x1 , y1 , z1 ) and P2 =
(x2 , y2 , z2 ) is
p
d := (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
Indeed, assume for simplicity P2 = O, the origin. Let us move from the origin to P = P1 , as
described above. Then |OQ| = |x1 |, |QR| = |y1 |, and the triangle OQR has right angle Q. By
the Pythagorean theorem, |OR|2 = |OQ|2 + |QR|2 = |x1 |2 + |y1 |2 . Similarly, the triangle ORA has
right angle R, and |RP | = |z1 |. So |OA|2 = |OR|2 + |RA|2 = |x1 |2 + |y1 |2 + |z1 |2 = x21 + y12 + z12 .
Example. The distance between (2, −1, 6) and (3, −3, 8) is
p
√
(2 − 3)2 + (−1 − (−3))2 + (6 − 8)2 = 1 + 4 + 4 = 3.
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Equations of spheres. A sphere with radius r and center P0 (x0 , y0 , z0 ) is the surface which
consists of all points P such that the distance from P to P0 is r. If P = (x, y, z), then its equation
is
p
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r ⇔ (x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r2 .
This is similar to the equation of the circle on R2 with center (x0 , y0 ) and radius r: (x − x0 )2 +
(y − y0 )2 = r2 .
Example. The unit sphere (i.e. centered at the origin with radius 1) has the equation x2 +
2
y + z 2 = 1.
Example. x2 − 2x + y 2 + 4y + z 2 = 5 also represents a sphere: let us complete the squares,
√ 2
x2 − 2x + 1 + y 2 + 4y + 4 + z 2 = 10 ⇔ (x − 1)2 + (y + 2)2 + z 2 = 10 .
√
This is a sphere with radius 10 centered at (1, −2, 0).
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