CHMA11H3 Lecture Notes - Lecture 5: Nspace, Mandala 3, Global Positioning System
Notation 1
Numbers
1. N={1,2,3,4,· · ·} is the set of natural numbers.
2. Z={0,±1,±2,±3,±4,· · ·} is the set of the integers.
3. Q={p
qp, q ∈Zand q6= 0}is the set of the rational numbers.
4. Q={x∈Rx6∈ Q}is the set of all irrational numbers.
5. Ris the set of all real numbers.
Vectors in the Euclidean n-space
DEFINITION: If nis a positive integer, the Euclidean n-space, Rn,is the collection of
all ordered n-tuples of real numbers.
There are two types of n-tuples in Rn:
(a1, a2,···, an) where ai’s are real numbers and it denotes a point.
[a1, a2,···, an] where ai’s are real numbers and it denotes a vector.
The zero vector in Rnis the vector containing zeroes in all of its components: 0= [0,0,···,0].
Geometric Interpretation of Vectors.
In R
In R2:
In R3
2c
2011 by Sophie Chrysostomou
We can generalize and say that the vector a= [a1, a2,···, an], in its standard position, is the
arrow that starts at the origin (0,0,···,0) and ends at the point (a1, a2,···, an) .
A vector of the same length and direction as a, is the same vector atranslated to another
position in Rn.
Vector Algebra in R:Let v= [v1, v2,···, vn] and w= [w1, w2,···, wn] be in Rnand
r∈R. We define
1. Vector Addition: v +w= [v1+w1, v2+w2,···, vn+wn].
2. Vector Subtraction: v −w= [v1−w1, v2−w2,···, vn−wn].
3. Scalar Multiplication: rv= [rv1, rv2,···, rvn].
Geometric Interpretation
4
GGG
w
GGGGGG
u
GG
GG
GG
G
G
GG
vrGGGG
vsGGG
v
GG )
v
GG
3c
2011 by Sophie Chrysostomou
36
CHMA11H3 Full Course Notes
Verified Note
36 documents
Document Summary
If n is a positive integer, the euclidean n-space, rn, is the collection of. There are two types of n-tuples in rn : (a1, a2, , an) where ai"s are real numbers and it denotes a point. [a1, a2, , an] where ai"s are real numbers and it denotes a vector. The zero vector in rn is the vector containing zeroes in all of its components: 0= [0, 0, , 0]. A vector of the same length and direction as a, is the same vector a translated to another position in rn. Vector algebra in r: r r. we de ne. Definition: parallel vectors we say that two nonzero vectors v and w are parallel and we denote it by vkw, if v = rw for some real number r. 0 < r < 1 r > 1. 1 < r < 0 r < 1. For all v, u, w rn and all r, s, r,
