11.2 Vectors in Space

A three-dimensional coordinate system consists of: 3 axes: x-axis, y-axis and z-axis 3 coordinate planes: xy-plane, xz-plane and yz-plane 8 octants.

( , , )P x y z

Each point is represented by an ordered triple

x

z

y

1 2 3, ,v v vv

Each vector is represented by

Coordinates in Space

Midpoint Formula:

Distance Formula:

1 2 1 2 1 2, ,2 2 2

x x y y z z

Given 2 points in the space with coordinates

1 1 1 2 2 2( , , ) ( , , )x y z and x y z

2 2 22 1 2 1 2 1( ) ( ) ( )d x x y y z z

2 2 2 20 0 0( ) ( ) ( )x x y y z z r

Standard equation of a sphere of radius r, centered at

0 0 0( , , )x y z is

1) Find the standard equation of the sphere whose endpoints of a diameter are (2,1,1) , (4, 5,3)

2 2 2 2 10 12 2 0x y z x y z

2) Find the center and radius of the sphere:

2 2 2 2 10 12 2x y z x y z

3) Describe the solid satisfying the condition:

Examples

and

1,0,0i

If 0v then v is a zero vector : 0 0,0,0

are called the standard unit vectors.

0,1,0j

The magnitude of 1 2 3, ,v v vv is:

2 2 21 2 3v v v v

If 1v then v is a unit vector.

0,0,1k

Vectors

1 2 3 1 2 3Let , , , , , , a scalar.u u u v v v k u v

1 1 2 2 3 3, ,u v u v u v u vVector sum:

1 1 2 2 3 3, ,u v u v u v u vVector difference

Scalar Multiplication: 1 2 3, ,k ku ku kuu

Negative (opposite): 1 2 31 , ,u u u u u

Vector v is parallel to u if and only if v = ku for some k.

Vector Operations

1) Find the unit vector in the direction of v

6,0,8v

(4, 2,7), ( 2,0,3), (7, 3,9)

2) Determine whether the points are collinear:

(1,1,3), (9, 1, 2), (11,2, 9), (3,4, 4)

3) Show that the following points form the vertices of a parallelogram:

Examples

(Normalize v)

1 2 3v v v v i j k

v is called a linear combination of i, j and k

Standard unit vector notation

Unit vector in the direction of v is given by

This unit vector is called the normalized form of v .

v

v

Linear Combination