COMPONENT FORM OF A VECTOR
Quantities (such as force, velocity, and
acceleration) involve magnitude and
direction. A directed line segment is used to
representuthese
uur quantities. The directed line
segment PQ has an initial point P and
terminal point Q, and itsulength
(or
uur
magnitude) is denoted PQ .
COMPONENT FORM OF A VECTOR
Directed line segments that have the same
length and direction are equivalent. The set
of all directed line segments that are
to a given directed line segment
uequivalent
uur
PQ is a vector in the plane.
VECTOR REPRESENTATION BY DIRECTED LINE SEGMENTS
Let v be represented by the directed line segment from
(0,0) to (3,2) and let u be represented by the directed line
segment from (1,2) to (4,4). Show that v and u are
equivalent.
AN INTRODUCTION TO VECTORS
If a vector starts at ( x1, y1 ) and terminates
at ( x2, y2 ), then its components are
< x2 – x1, y2 – y1 >
The magnitude v is the length of the
vector.
Find the component form for each vector. Find the
magnitude of the vector.
a. Initial point of (2, 3) and terminal point of (7, 6)
b. Initial point of (3, 1) and terminal point of (2, - 3)
DEFINITIONS OF VECTOR ADDITION AND
SCALAR MULTIPLICATION
Let u = u1,u2 and v = v1,v2 be vectors and let c
be a scalar.
1. The vector sum of u and v is the vector u + v = u1 + v1,u2 + v2
2. The scalar multiple of c and u is the vector cu = cu1,cu2
3. The negative of v is the vector
-v = (-1)v = -v1,-v2
4. The difference of u and v is
u - v = u1 - v1,u2 - v2
VECTOR OPERATIONS
Given v = -2,5 and w = 3, 4 , find each of the
vectors.
a. ½v
b. w – v
c. v + 2w
PROPERTIES OF VECTOR OPERATIONS
Let u, v, and w be vectors in the plane and let c and d be scalars.
1. u + v = v + w
2. (u + v) + w = u + (v + w)
3. u + 0 = u
4. u + (-u) = 0
5. c(du) = (cd)u
6. (c + d)u = cu + du
7. c(u + v) = cu + cv
8. l(u) = u, 0(u) = 0
A car travels with a velocity vector given by:
v(t) = t ,e +1
2
t
where t is measured in seconds, and the vector components are
measured in feet.If the initial position of the car is:
r(0) = 1, 3
find the position of the car after 1 second.