04/10/23 6.4: Coordinates in 3 Dimensions

6.4: Coordinates in Three Dimensions

What is the sum of the 2 real solutions to the equation x = 6 –

x2?

a. 6b. 2c. 1d. -1e. -6

04/10/23 6.4: Coordinates in 3 Dimensions

Three Dimensional Coordinates

1. x-axis, y-axis, z-axis.2. (x,y,z) ordered triples.

04/10/23 6.4: Coordinates in 3 Dimensions

3 Dimensional Coordinate System

x

y

z

04/10/23 6.4: Coordinates in 3 Dimensions

Plot (3,2,5)

04/10/23 6.4: Coordinates in 3 Dimensions

Plot (-4,-1,2)

04/10/23 6.4: Coordinates in 3 Dimensions

Plot (0,5,-4)

04/10/23 6.4: Coordinates in 3 Dimensions

Coordinate PlanesThere are 3 coordinate planes in a 3

dimensional coordinate system. They are the xy plane, the yz plane and the xz plane.

In the xy plane, the z coordinate of all points is 0. In the yz plane the x coordinate of all points is 0. In the xz plane the y coordinate of all points is 0.

04/10/23 6.4: Coordinates in 3 Dimensions

OctantsThe xy, yz and xz planes split

space into 8 regions called octants.

The only octant that has a specific name is the region in which all 3 coordinates are positive. This octant is called the first octant.

04/10/23 6.4: Coordinates in 3 Dimensions

OctantsThe other 7 octants are called

out by location i.e. top back right or bottom front left.

04/10/23 6.4: Coordinates in 3 Dimensions

Give the location (octant) of:

(-4,9,8)

Back, right, top

(3,9,0)

xy plane

(-4, -8,-1)

Back, left, bottom

04/10/23 6.4: Coordinates in 3 Dimensions

Give the coordinates of a point:

In the bottom back right octant.

In the top front left octant

In the xz coordinate plane.

04/10/23 6.4: Coordinates in 3 Dimensions

Distance Formula in Three Dimensions

The distance d between the points (x1,y1,z1) and (x2,y2,z2) is given by

d=

2)1

z2

(z2)1

y2

(y2)1

x2

(x

04/10/23 6.4: Coordinates in 3 Dimensions

Calculate the distance between (3,6,9) and (2,4,-8).

Give the coordinates of 2 points that are 10 units apart in the 3-d coordinate system.

04/10/23 6.4: Coordinates in 3 Dimensions

3-D Midpoint FormulaIf A(x1,y1,z1) and B(x2,y2,z2), then

the coordinates of the midpoint, M, of AB are given by the formula:

M = x1+x2 , y1+y2 , z1+z2

2 2 2

04/10/23 6.4: Coordinates in 3 Dimensions

What are the coordinates of the midpoint of the segment determined by (-2,4,3) and

(8, 12,-9)?

04/10/23 6.4: Coordinates in 3 Dimensions

Assignment pages 399-400, # 14 - 48(evens)