6.3 Vectors in the Plane
Vector
 A directed line segment with initial point P

and terminal point Q PQ
 Magnitude (length) is denoted PQ
 The set of all directed line segments that are
equivalent to a given directed line segment
PQ is a vector v in the plane, v =PQ
 Vectors are denoted by lowercase boldface
letters, usually u, v, and w
 A vector in standard position has it’s
initial point at the origin
Component Form of Vectors
With initial point P=(p1, p2) and terminal
point Q=(q1, q2)

PQ  q1  p1 , q2  p2  v1 , v2  v
The magnitude (length) of v is
v  (q1  p1 ) 2  (q2  p2 ) 2  v12  v22
If v  1 then v is a unit vector
If v  0 then v is the zero vector
Example 1: Find the component form
and magnitude of the vector with
initial point (-2, 7) and terminal point
(4, -5)
Vector addition and subtraction
Let u  u1 , u2 and v  v1 , v2 and let k be scalar
u  v  u1  v1 , u2  v2
ku  ku1 , ku2
 v   v1 ,v2
u - v  u1  v1 , u2  v2
Example 2:
If u = 11, 3 and v  2, 7
Find 2v – u
Find 3u+5v
Properties of vectors
 Let u, v, and w be vectors and c and d
be scalars.
1.
2.
3.
4.
5.
u+v = v+u
(u+v)+w=u+(v+w)
u+0=u
u+(-u)=0
c(du)=(cd)u
6. (c+d)u=cu+du
7. c(u+v)=cu+cv
8. 1(u)=u, 0(u)=0
9. ||cv||=|c| ||v||
Unit Vectors
v  1 
  v
u = unit vector =
v  v 
u is a scalar multiple of v with a
length of 1 and the same direction as
v
u is called a unit vector in the
direction of v
Standard unit vectors

i  1,0 and j  0,1
 A linear combination
of vectors i and j is
the sum v1i + v2j
where v  v1 , v2
y
j
i
x
Example 3: Find a unit vector in the
direction of the given vector
w = 4i - 3j
Direction angles
 If u is a unit vector and θ is the angle from the
positive x-axis to u, then u lies on the unit circle
and u  x, y  cos  , sin   cos i  sin j
For any unit vecto r
v  ai  bj  v cos i  v sin j
b
tan  
a
Example 4: Find the component form
of v given the magnitude and the
angle
v  9   90
Assignment
Page 453 #3-13odd, 21-35odd, 4357odd