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Vectors in Euclidean Spaces
Vectors.
- Economists usually work in the vector space R n . A point in this space is called
a vector, and is typically defined by its rectangular coordinates.
n
. We define this vector by its n coordinates, v 1 , v 2 ,... , vn.
It is common to write v = (v 1 , v 2 ,... , vn) or to display a vector as a column
matrix:
v =
v 1
v 2
. . .
vn
- It is common to distinguish between locations and dispacements by writing a
location as a row vector and a displacement as a column vector. However, we
can use the same algebraic operations to work with each.
- A vector can be also be defined by its origin and end points.
- Suppose the vector v links the point P = (p 1 ,... , pn) to the point Q = (q 1 ,... , qn)
in R n
. Then v = Q − P , i.e. vi = pi − qi, ∀i ∈ { 1 , 2 ,... , n}.
v
P
Q
Figure 1: The displacement (5, −3)
Addition.
- Given two vectors u and v, with coordinates, we add them like so:
u + v =
u 1
u 2
. . .
un
v 1
v 2
. . .
vn
u 1 + v 1
u 2 + v 2
. . .
un + vn
v
v
u
u + v
v 1 u 1 u 1 +^ v 1
u 2
v 2
u 2 + v 2
Figure 2: Vector Addition
Scalar Multiplication.
- We can also multiply vectors by scalars. Suppose λ ∈ R and v ∈ R
n
. Scalar
multiplication gives a vector λv ∈ R
n , defined by
λv =
λv 1
λv 2
. . .
λvn
u
− 2 u
v
1 2
w v
3 w
Figure 3: Scalar multiplication
Subtraction.
- The difference of two vectors, say u − v, is the sum of the vector u with the
vector −v = (−1)v.
u − v =
u 1
u 2
. . .
un
v 1
v 2
. . .
vn
u 1 − v 1
u 2 − v 2
. . .
un − vn
Laws of Vector Algebra.
- Let λ, β ∈ R and u, v, w ∈ R
n
. Then the following algebraic properties of
vectors hold.
u · v = v · u
u · (v + w) = u · v + u · w
Length and Inner Product.
Definition. The norm or length of a vector u is the real number, denoted ‖u‖, given by
‖u‖ =
u
2 1 +^ u
2 2 +^ · · ·^ +^ u
2 n.^ N
- Using our definition of the inner product we can also write this as
‖u‖ =
u.u
- The norm of a vector is always positive unless the vector is the zero vector, in
which case the norm is zero.
- The distance between two vectors u, v ∈ R n is calculated as
‖u − v‖ =
(u 1 − v 1 ) 2
- Note that for any λ ∈ R and u ∈ R
n :
‖λu‖ = |λ|‖u‖.
u · v = ‖u‖‖v‖ cos θ
where θ is the angle between the vectors u and v.
- Using the properties of the cosine we get the following result.
Theorem 1. The angle between vectors u and v in R n is
- acute, if u · v > 0 ,
- obtuse, if u · v < 0 ,
- right, if u · v = 0.
Definition. Let v be a vector. The vector w which points in the same direction as v,
but has length 1 is called the unit vector in the direction of v (or simply the direction
of v). It is given by
w =
v
‖v‖
. N
u
v
‖u‖ cos θ
θ
Figure 5: The angle between two vectors u and v.
Definition. Two vectors u, v ∈ R n are orthogonal if u · v = 0. N
- This definition implies the zero vector is orthogonal to any vector.
Definition. Two vectors u, v ∈ R are orthonormal if they are orthogonal and are unit
vectors. N
Theorem 2 (Triangle Inequality). For any two vectors u, v ∈ R
n ,
‖u + v‖ ≤ ‖u‖ + ‖v‖.
Theorem 3 (Triangle Inequality Variant). For any two vectors u, v ∈ R
n ,
|‖u‖ − ‖v‖| ≤ ‖u − v‖.
- There are three basic properties of Euclidean length for any vectors u and v and
scalar λ:
- ‖u‖ ≥ 0 and ‖u‖ = 0 only when u = 0,
- ‖λu‖ = |λ|‖u‖,
- ‖u + v‖ ≤ ‖u‖ + ‖v‖.
Any assignment of a real number to a vector satisfying these properties is called
a norm (see sections 29.4 and 27 of S&B if interested).
Projections.
n
. We want to find the vector projection of the vector u in the
direction of v.
- Denote the projection of u on v by Pv (u). We can see from the diagram that the
length of Pv (u) (called the scalar projection of vector u on v) is given by
‖Pv (u)‖ = ‖u‖ cos θ.
p + tv v
p
Figure 7: Parametric line ` in R 2
Two-dimensional Planes.
- We saw that a line – a one-dimensional object – can be described using only one
parameter.
- A plane is two-dimensional, so we need two parameters.
- Consider a plane P in R
3 and let u and v be two vectors in P that point in
different directions:
x 1
x 2
x 3
u
v
Figure 8: A plane P through the origin in R 3
- We can move from the origin in direction u, v or any combination of the two.
So, for any scalars s and t, the vector su + tv also lies in the plane P.
- Thus any plane P through the origin, in a vector space R n (n > 2 ), can be
defined in its parametric form as
P = {x ∈ R
n | x = su + tv, s, t ∈ R}
- But what if the plane does not pass through the origin?
- Suppose the plane does not pass throught the origin.
- We can move from point p in the plane in direction u, v or any combination of
the two. Thus the vector p + su + tv also lies in the plane P.
x 1
x 2
x 3
p
u v
Figure 9: A plane P not through the origin in R 3
- So any plane P through the point p, in a vector space R n (n > 2 ), can be defined
in its parametric form as
P = {x ∈ R
n | x = p + su + tv, s, t ∈ R}
- A point x ∈ R n belongs to the plane P iff there exist two scalars s and t such
that x = p + su + tv. Equivalently, the vector x − p must be a linear combination
of the vectors u and v.
- As two points uniquely determine a line, three distinct (non-collinear) points P ,
Q and R uniquely determine a plane.
- Let u = Q − P and v = R − P. We can picture these as displacement
vectors from P :
x 1
x 2
x 3
P
u
Q
v
R
Figure 10: A plane P not through the origin in R 3
- Remember, we need u and v to be nonparallel to for them to uniquely determine
a plane.
Hyperplanes.
2 can be written as
a 1 x 1 + a 2 x 2 = d
and a plane in R
3 can be written in point-normal form as
a 1 x 1 + a 2 x 2 + a 3 x 3 = d.
- Generalizing, a hyperplane in R n can be written in point-normal form as
a 1 x 1 + a 2 x 2 + · · · + anxn = d,
where (a 1 , a 2 ,... , an) is a normal.
- The set of vectors in the hyperplane have tail at (0,... , 0 , d/an) and are
perpindicular to the normal vector to the hyperplane.
Example 1.
- An economic application you have probably seen deals with commodity spaces.
x = (x 1 , x 2 ,... , xn)
of nonnegative quantities of n commodities is called a commodity bundle.
The set of all commodity bundles is the set
{(x 1 ,... , xn) | x 1 ≥ 0 ,... , xn ≥ 0 }
and is called a commodity space.
- Let pi > 0 be the price of commodity i. The cost of buying bundle x is
p 1 x 1 + p 2 x 2 + · · · + pnxn = p · x.
A consumer with income I can purchase only bundles x for which p·x ≤ I.
This subset of the commodity space is the consumer’s budget set.
- • The budget set is bounded above by the hyperplane p·x = I, whose normal
vector is the price vector p.
x 1
x 2
p = (p 1 , p 2 )
p 1 x 1 + p 2 x 2 = I
Figure 12: A consumer’s budget set, p · x ≤ I, in commodity space.
