Vectors in three dimensions

Vectors in Three-Dimensional Space

​Al Hosn School

​

Express algebrically and operate with vectors in space

2.

Plot points and vectors in the three-dimensional coordinate system

1.

Learning Outcomes

Coordinates in Three Dimensions

Three-dimensional coordinate system is system that is used to identify points in space.
When we deal with Three-dimensional space, We add third axis called the Z-axis that passes through the origin and is perpendicular to both the x- and y-axes.

Example 1

a. (4, 6, 2)

​b. (-2, 4, -5)

Multiple Choice

1A. (-3, -4, 2) note green is the Y-axis

Multiple Choice

1B. (3, 2, -3) "note green is the Y-axis"

Distance and Midpoint formula

Example 2

A tour of the Sierra Madre Mountains lets guests experience nature by zip-lining from one platform to another over the scenic surroundings. Two platforms that are connected by a zip-line are represented by the coordinates (10, 12, 50) and (70, 92, 30), where the coordinates are given in meters.

b. An additional platform is to be built halfway between the existing platforms. Find the coordinates of the new platform.

a. Find the length of the zip-line needed to connect the two platforms.

Multiple Choice

(-4, 10, 4), (1, 0, 9) Find the length and the midpoint

• length= 12.2

  midpoint= (-1.5, 5, 6.5)

• length= 4.6

  midpoint= (1, -2, 6.5)

• length= 10.2

  midpoint= (1.5, 5, -6.5)

Multiple Choice

(8, 3, 4), (-4, -7, 5) Find the length and the midpoint

• length= 8.3

  midpoint= (2, -4, 6.5)

• length= 12.7

  midpoint= (2, 4, -4.5)

• length= 15.6

  midpoint= (2, -2, 4.5)

Locate a vector in space

vectors are i = 〈1, 0, 0〉, j = 〈0, 1, 0〉, and k = 〈0, 0, 1〉

So the i stands for the x-axis, j for the y-axis, and the k is for the z-axis