13.1 Three-Dimensional Coordinate Systems

1. Terms

(1) Three-dimensional rectangular(Cartesian) coordinate system : A coordinate system on a space(threedimensional) in which a point is addressed uniquely by three numerical coordinates with respect tothe three perpendicular directed lines called x, y, and z axes. The three axes meet at a single pointand the point is called the origin, often denoted by O.

(2) Right hand rule : In the three-dimensional rectangular coordinate system, the x, y, and z axesare conventionally put in the direction to which the thumb, index, and middle fingers of a righthand(when the fingers are perpendicular to each other) are pointing.

(3) Rectangular(Cartesian) coordinates of a point in a space : When P is a point in a rectangular coor-dinate system and A, B, and C are perpendicular feet from P to x, y, and z axes respectively, thethree signed distances from the origin to A, B, and C are called x, y, and z coordinates of P . Thethree ordered numbers form the rectangular coordinates of P .

(4) Coordinate planes :xy plane : the plane containing x and y axes. Points in xy plane have zero z coordinate.

yz plane : the plane containing y and z axes. Points in yz plane have zero x coordinate.

zx plane : the plane containing z and x axes. Points in zx plane have zero y coordinate.

(5) Octants : A space is divided into eight octants by the coordinate planes. The octant bounded byall three positive coordinates is referred to as the first octant. There is no generally used namingconvention for the other seven octants.

2. Example Draw all points in the space satisfying each equation.

(a) x = 1 (b) y = 3 (c) z =

(d) x = y (e) y = z (f) z = y2

(a) (b) (c) (d) (e) (f)

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3. Formula (Distance between two points)

When P1(x1, y1, z1) and P2(x2, y2, z2) are distinct points in arectangular coordinate system, the distance between the twopoints is

P1P2 =(x1 x2)2 + (y1 y2)2 + (z1 z2)2

4. Example Determine if the triangle4ABC is an isosceles, a right triangle, both, or neither.

(a) A(5, 5, 1), B(3, 3, 2), C(1, 1, 4)

(b) A(3,4, 1), B(5,3, 0), C(6,7, 4)

5. Example (a) Find an equation for a point P(x, y, z) whose distance from O(0, 0, 0) equals r(> 0).

(b) Find an equation for a point P(x, y, z) whose distance from C(a, b, c) equals r(> 0).

Answers : (a)

(x 0)2 + (y 0)2 + (z 0)2 = r x2 + y2 + z2 = r2

(b)

(x a)2 + (y b)2 + (z c)2 = r (x a)2 + (y b)2 + (z c)2 = r2

6. Definition (Sphere)The collection of all points in space whose distances from a fixedpoint C(a, b, c) is equal to a positive number r is called a sphere.We call the point C the center and the number r the radius of thesphere. We call

(x a)2 + (y b)2 + (z c)2 = r2

the standard equation of a sphere centered at (a, b, c) with radiusr(> 0).

7. Example (a) Find an equation of the sphere centered at (1, 2, 4) with radius 3.

(b) Determine if the equation is a sphere :

(b1) x2 + y2 + z2 = 6x+ 4y + 10z(b2) 2x2 + 2y2 + 2z2 + 4y 2z + 2 = 0

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