PRE-CALCULUS NOTES
SECTION 6.3 VECTORS IN THE PLANE
A.) Vectors – quantities that involve both magnitude and direction. (i.e. Force)
- directed line segments
- usually denoted by u, v, or w
- have an initial and terminal point
Standard Position of a vector – initial point is at the origin.
Length is denoted by v or u (vector v = PQ , u = RS )
4
S
3
2
1
R
-2
P
-1
Q
2
4
6
8
10
u  RS  (5  1)2  (4  1)2  25  5
-2
v  PQ  (3  0)2  (1  0)2  10
-3
-4
-5
B.) Component Form of a Vector – have 2 components – Horizontal and Vertical
The component Form of a Vector with initial point P   p1 , p2  and terminal point
Q   q1 , q2  is PQ  q1  p1 , q2  p2  v1 , v2  v
The length or magnitude of v is given by
v  (q1  p1 )2  (q2  p2 ) 2  v1  v2
2
2
If v  1 , then v is a unit vector. v  0 if and only if v is the 0 vector = 0,0 .
( initial and terminal point origin )
Def.: Equivalent Vectors – have the same components ( same direction and magnitude ).
NOTE: Don’t have to have the same initial and terminal points.
2.) Show that v  (3,3) to (0, 5) is equivalent to u  (0, 4) to ( 3, 4) .
12.) Find the Component Form and magnitude of vector v.
Initial Point =  3,11 , Terminal Point =  9, 40 
C.) Vector Operations – Scalar Multiplication and Vector Addition
1.) Scalar Multiplication – scalar multiple of k times u is
ku  k u1 , u2  ku1 , ku2 makes vector longer or shorter
( If k is positive, ku has the same direction as u. )
( If k is negative, ku has the opposite direction as u. )
2.) Vector Addition – sum of u and v is the sum of their vectors.
u  v  u1  v1 , u2  v2
20.) u  5,3 and v  4,0
b.) find u  v
a.) find u  v
c.) find 2u  3v
d.) find v  4u
How do you express these operations GRAPHICALLY?
4
initial point of 1 vector coincides with terminal point of
another vector ( Parallelogram Law )
3
2
1
-4
-3
-2
-1
1
2
3
4
-1
u + v is called the RESULTANT – diagonal of the
parallelogram having u and v as its adjacent sides
-2
Ex.) Graphically start at the origin. u  1, 2 , v  0, 4
uv
v  2u
7
4
6
3
5
2
4
3
1
2
-4
1
-3
-2
-1
1
-1
-2
-1
1
-1
2
-2
2
3
4
Properties of Vector Addition and Scalar Multiplication
5.) c(du )  (cd )u
1.) u  v  v  u
2.) (u  v)  w  u  (v  w)
6.) (c  d )u  cu  du
3.) u  0  u
7.) c(u  v )  cu  cv
4.) u  (u )  0
8.) 1(u )  u , 0(u )  0
9.) cv  c v
D.) Unit Vectors – vector u which has length 1 and the same direction as a given nonzero vector
v. ( u is a scalar multiple of v )
v  1 
  v
Unit Vector = u 
v  v 
28.) find unit vector of v  3, 4
1.) Standard Unit Vectors are 1, 0 and 0,1 which represent horizontal and vertical
components denoted by i = 1,0 and j = 0,1 . These standard unit vectors can be used to
represent any vector v  v1 , v2 :
v  v1 , v2  v1 1,0  v2 0,1
v  v1 i + v2 j is called linear combination of vectors i and j
2.) Linear Combination – vector sum v1 i + v2 j
- any vector in the plane can be expressed as a linear combination of the standard
unit vectors i and j.
44.) Find the linear combination of v given: initial point =  6, 4  and terminal point =  0,1 .
38.) Find the vector v with v  10 and the same direction as u  2i – 3j. (Hint: find unit vector
and give it a magnitude of 10.)
50.) Find component form of v  2u  2w given u = 2i – j and w = i + 2j.
E.) Direction Angles – angle  that represents direction of vector in the plane from the positive
x-axis to the vector.
- Any point on that circle can be represented by  cos ,sin  
-
u  cos ,sin   cos  i +  sin   j
Any vector can be represented in terms of its unit vector.
v  v u where u = unit vector
so v  v cos ,sin  v  cos  i + v  sin   j
To find direction angle  , recall:
v sin 
sin 
tan  

cos
v cos 
56.) Find the magnitude and direction angle of v  12i + 15j.
62.) Find the component form of v given: v  3 and v is in the direction of 3i + 4j.
66.) Find the component form of the sum of u and v.
u  35
 u  25
v  50
 v  120