6-3: VECTORS IN THE PLANE
Various quantities in geometry and physics such as area, time and
temperature can be measured in a single real number.
However, quantities such as force and velocity have both a magnitude and
direction that need to be characterized by more than a real number
Vectors
* directed line segments with the same magnitude and direction are
equivalent
-they have the same distance and same slopes!
* represented by bold-faced, lowercase letters
Example:
u is the line segment from P=(0,0) to Q=(3,2) and v is the line segment from
R=(1,2) to S=(4,4). Show that u=v.
Determine magnitude: (distance formula)
Determine direction: (slope)
Component Form of a Vector
v1 ,v2
Find the component form and magnitude of a vector v that has initial point
(4, -7) and terminal point (-1, 5).
Solution:
v1  change in the x values
If v=
 5,12
v2  change
, the magnitude of v is
v  (5) 2  (12) 2  169  13
in y values
Vector Operations
Let v=  2,5 and w= 3,4 , and find each of the following vectors.
a. 2v
b. w – v
c. v + 2w
d. 2v - 3w
Unit Vectors
*determine unit vectors by dividing v by its magnitude
Find a unit vector in the direction of v=  2,5 .
v

v
 2,5
(2) 2  (5) 2
v
.

Linear Combinations of Unit Vectors
Another way to represent vectors in equation form
 2,5 = -2i + 5j
Operations with linear combinations work just like combining like terms
Example:
u= -3i + 8j and v= 2i – j find 2u – 3v.