3D Coordinate System
Graph P(2, 4, -2) and Q(-3, 0, 5)

A vector is a quantity specified by magnitude and direction.
A scalar is a quantity specified by magnitude only.
Geometric
Vectors are represented by an arrow.
Operations

**** Two vectors are parallel if they are scalar multiples of each other.***** is a nonzero constant.
If c > 0, the vectors go in the same direction.
If c < 0, the vectors go in the opposite direction.
is the unit vector in the positive x direction.
is the unit vector in the positive y direction.

ex.
The magnitude of v,
A unit vector is a vector of length 1.
.

A vector is a product of magnitude and direction.
Direction is a UNIT vector.
Algebraic addition: ex. Find the horizontal and vertical components of if has magnitude 3 and makes an angle of 20 degrees with the horizontal.

ex. From an old Engineering test:
Vector A has a magnitude of 24.3 N, begins at the point (0,0) and goes through the point (-2,4). Vector B has a magnitude of 13.8
N, begins at point (1, -2), and goes through the point (2, -1). Find the resultant vector A + B.





Do: 1. Force A has a magnitude of 20 N and points in a direction of 150 degrees from the positive x-axis. Force B has a magnitude of 6 N at an angle of 270 degrees. The resultant force A + B is:
2. Suppose P(1, 3) and Q(5, -1). Find . a.

A

B
 
10 3, 16 and QP

4,

4 b.

A

B
 
10 3, 16 and QP
 
4, 4 c.

A

B
 
10 3, 4 and QP

4,

4 d
.

A

B
 
10 3, 4 and QP
 
4, 4

3D
is the unit vector in the positive z direction. ex. P(7, 1, 1) Q(0, 2, 3)




 r u

2,

1, 4 , r v

1, 3, 4 , ur w
Are any of these vectors parallel?
 
1,
1
2
,

2
Do: 1. Find a vector of length 3 in the same direction as .
. c.
d
.
2. Find point B if A(1, 3), a.
3
26
,
9
26
,
12
26
, B
  b
.
3
26
,
9
26
,
12
26
, B

7,11

3, 9, 12 ,
3, 9, 12 ,
B
B

6, 8


7,11
