1.1 – 1.2
The Geometry and Algebra
of Vectors
Vectors in the Plane
 Quantities that have magnitude but not direction are
called scalars.
Ex: Area, volume, temperature, time, etc.
 Quantities such as force, acceleration or velocity that
have direction as well as magnitude are represented by
directed line segments, called vectors.
AB
Initial A
point
(tail)
B terminal
point
(head)
 The length of the vector is called
the magnitude and is denoted by
AB
 Vectors are equivalent if they have the same
length and direction (same slope).
y
 A vector is in standard position if
the initial point is at the origin.
 v1 , v2 
 The component form of this vector
is:
v  v1 , v2 or v  v1, v2 
 If
x
 v1 
or v   
v2 
P  (c, d ) and Q  (a, b)
(c,d)
P
(a,b)
are initial and terminal points of a vector,
Q
x
then the component form of
PQ is: v  a  c, b  d 
v
(a-c, b-d)
Example
(-3,4)
P
(-5,2)
Q
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1 0
v -1
-2
(-2,-2)
-3
-4
-5
-6
The component form of
PQ is: v   2,2
The magnitude is
1 2 3 4 5 6
v 
 2
 8
2 2
2
  2 
2
The magnitude of
v  v1, v2 
is: v 
If
v 0
then v is a zero vector:
If
v 1
then v is a unit vector.
e2
6
5
4
3
2
1
e1  1,0
v12  v22
0  0,0
and
e2  0,1
are called the standard unit vectors.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1 e1
-2
-3
-4
-5
-6
Vector Operations
Let u  u1 , u2 , v  v1 , v2 , k a scalar (real number).
Vector sum:
u  v  u1  v1 , u2  v2
Scalar Multiplication:
ku  ku1 , ku2
Negative (opposite):
u   1 u  u1 , u2
Vector difference
u  v  u1  v1 , u2  v2
The Parallelogram Rule
u
v
(The Head-to-Tail Rule)
u+v
u + v is the resultant vector.
v
u
u
v
u-v
v
u
u - v is the resultant vector.
Linear Combination
A vector v is called a linear combinatio n of vectors
v1 , v 2 ,..., v n if there are scalars c1 , c2 ,..., cn such that
v  c1v1  c2 v 2  ...  cn v n .
Coordinates in Space
A three-dimensional coordinate system consists of:
 3 axes: x-axis, y-axis and z-axis
 3 coordinate planes: xy-plane, xz-plane and yz-plane
 8 octants.
Each point is represented by an ordered triple
Each vector is represented by
P ( x, y , z )
z
v  v1 , v2 , v3
v  v1 , v2 , v3 
x
y
Vectors in Space
The magnitude of
v  v1 , v2 , v3 
is:
v  v12  v22  v32
If
v 0
then v is a zero vector : 0  0, 0, 0
If
v 1
then v is a unit vector.
e1  1,0,0
e2  0,1,0
and
e3  0,0,1
are called the standard unit vectors.
Vectors Operations
Let u  u1 , u2 , u3 , v  v1 , v2 , v3 , k a scalar.
Vector sum:
u  v  u1  v1 , u2  v2 , u3  v3
Scalar Multiplication:
Negative (opposite):
Vector difference
ku  ku1 , ku2 , ku3
u   1 u  u1 , u2 , u3
u  v  u1  v1 , u2  v2 , u3  v3
Vector v is parallel to u if and only if v = ku for some k.
n
n
Vectors in R and Z
The set of all vectors with n real-valued components is
denoted by Rn . Thus, a vector in Rn has the form
v  v1, v2 ,..., vn 
 R2 is the set of all vectors in the plane.
 R3 is the set of all vectors in three-dimensional space.
 Z3 is the set of all vectors in three-dimensional space
whose components are integers.
 Z3 is the set {0, 1, 2} with special operations (Integer
modulo 3)
Algebraic Properties of Vectors
1) Commutativ e :
2)
3)
4)
5)
6)
7)
vuuv
Associativ e : (u  v)  w  u  (v  w)
Additive Identitiy : u  0  u
Opposite :
u  (  u)  0
Distribu tive :
c(v  u)  cu  cv
Distribu tive :
( c  d ) u  c u  du
Multipli cative Identity : 1u  u
Definitions
Let u  [u1,u2 ,..., un ] and v  v1, v2 ,..., vn 
The dot product of u and v is defined by
u  v  u1v1  u2v2  ...  un vn
The dot product is also called scalar product.
(Read “u dot v”)
Two vectors u and v are orthogonal
 if they meet at a right angle.
 if and only if u ∙ v = 0.
The distance between vectors u and v is
defined by
d ( u  v)  u  v
Example
3, 4  5, 2   3 5   4  2   23
2, 3  3, 2   23   3 2  0
Properties
Let u, v, w be vectors
1.
2.
3.
4.
u  v  v u
u  ( v  w)  u  v  u  w
c(u  v)  cu  v  u  cv
0 v  0
5. v  v  v
2
Another form of the Dot Product:
u  v  u v cos
where  is the angle between two nonzero vectors u and v.
Example
Find the angle between vectors u and v:
u  2,3 , v  2,5
 uv 
11 
1 
  cos 
  cos 
 55.5

 13 29 
u v
1
Vector Components
u
w2
w1
 u v 
w1  projv u   2  v
 v 


w 2 = u - w1
v
Let u and v be nonzero vectors.

w1 is called the vector component of u along v
(or projection of u onto v), and is denoted by projvu

w2 is called the vector component of u orthogonal to v
Examples
u  2,1,3 ｖ   3,0,4
1. Write u as a linear combination of standard unit vectors.
2. Find u + v and 2u – 3v.
3. Are u and v parallel? orthogonal?
4. Find the angle between u and v.
5. Find the magnitude of v.
6. Normalize vector v.
7. Find the projection of u onto v.
8. Find the vector component of u orthogonal to v.
9. Find the projection of v onto u.
10. Find the distance between u and v.