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Summary Mathematics 2
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Summary of Mathematics 2 for BA2 Business Economics at the VUB.
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VECTORS IN SPACE,Chapter 2: Vectors
KEY TERMS:
Magnitude: The length of a vector and is denoted as ‖a‖ .
Direction: General line that is moving or pointing in.
1.1. Vectors
Vector: An object consisting of a magnitude and a direction. It can model a displacement.
Example:
2 (or more) vectors with the same magnitude and direction are equal.
example:
The model above model a displacement of one over and two up.
Denoting the vector that extends from ("! , "" ) &' ((! , (" ) by
"! − $!
! %
"" − $"
The “1 over, 2 up” could be written as !!""
Vector : #⃗ = '### ( as starting from the origin (0,0). From there *⃗ extends to (*! , *" ) and we
$
"
May refer to it as “the point *⃗” so that we call each of these ℝ .
{(+! , +" ⎹ +! , +" ∈ ℝ } {#!!! $ ⎹ +1 , +2 ∈ ℝ}
"
These definitions extend to higher dimensions. The vector starts at ($1, "2 , … , $& ) and ends at
("1 , "2 , … , "& ) is represented by this column:
(! − "!
- . 0
.
(% − "%
And 2 vectors are equal if they have the same representation.
⎧ 1& ⎫
ℝ' = . . 0 ⎹ 1& , … , 1' ∈ ℝ
⎨ . ⎬
⎩ 1' ⎭
1.2 Vector operations
Scalar multiplication:
Makes a vector longer or shorter, including possibly flipping it around.
,Addition:
Where *⃗ and 4
55⃗ represent displacement, the vector *⃗ + 4
55⃗ represents those displacements
combined. è the parallelogram rule for vector addition.
Subtraction:
*⃗ − 4
55⃗ = *⃗ + (−1)4
55⃗
8
99⃗ ;⃗
;⃗ − 8
99⃗
−8
99⃗
1.3 Lines
The line in ℝ" through (1,2) and (3,1) is comprised of the vectors in this set. (That is, it is
comprised of the endpoints of those vectors)
"
The vector associated with parameter t !&!" is the direction vector for the line. (Lines in
higher dimensions work the same way)
1.4 Plane in ℝ(
Need 1 base vector and 2 direction vectors.
If 2 vectors are the, they represent a line – They are colinear.
1.5 n-dimensional space
Can construct objects with k-dimensions
⎛ ".! ⎞ ⎛ (.! ⎞ ⎛ @.! ⎞ ⎛ B.! ⎞
⎜ + ?! ⎜ . ⎟ + ?" ⎜ . ⎟ + ⋯ + ?% ⎜ . ⎟
⎜ . ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
. . . .
⎝$& ⎠ ⎝ "& ⎠ ⎝6& ⎠ ⎝7& ⎠
Base vector
Line
Plane
k-dimension hyperplane
, 2. Length and angle measures
2.1. Lengths
Definition: The length of a vector *⃗ ∈ ℝ% is the square root of the sum of the square of its
components.
5⃗| = D(1)& + ⋯ + 1)' )
|1
&!
Example: length of 1&"2 is √1 + 4 + 9 = √14
&'
#(⃗
For a nonzero vector *⃗, the length of one vector with the same direction is .
|#(⃗|
5⃗ to unit length.
We say that this normalizes 1
2.2. Dot product
Definition: The dot product (or inner product or scalar product) of two n-components real
vectors is the linear combination of their components.
5H⃗ ∘ 51⃗ = H& 1& + H) 1) + ⋯ + H' 1'
Example: the dot product of two vectors, is a scalar, not a vector.
1 3
1
J K −3N = 3 − 3 − 4 = −4
∘ L
−1 4
9⃗ ∘ 9=⃗ = =)( + ⋯ + =)' is the square of the vector’s
The dot product of a vector with itself =
length.
Length of a vector in 2-dimensions -> Pythagoras
Length of a vector in 3-dimensions -> |1 5⃗)& + 51⃗)) + 51⃗)*
5⃗| = D1
(⃗
#
Normalizing vector:
|#
(⃗|
2.3 Triangle inequality
5⃗, *⃗ ∈ ℝ" ,
2.5 Theorem: for any O
|H
5⃗ + 51⃗| ≤ |H
5⃗| + |1
5⃗|
With equality if and only if one of the vectors is a nonnegative scalar multiple of the other one.
This is the source of the familiar saying, “The shortest distance between 2 points is in a
straight line.”
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