Euclidean Spaces. Euclidean Spaces. Chapter 10 -S&B
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1 Chapter 10 -S&B
2 The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane or Euclidean 2-space, R 2.
3 Three Dimensions and More 3 dimensional Euclidean space, R 3. Coordinate axis: the x-axis, the y-axis and the z-axis R 1 consists of single numbers; R 2 consists of ordered pairs of numbers and R n the euclidean n-space, refers to how many numbers are needed to describe each location (R 5 is (a, b, c, d, e))
4 Vectors We can think of n-tuples of number as locations (i.e. location in commodity space). We can also interpret n-tuples as displacements. We picture these displacements as arrows in R n. (2, 3) means: move 3 units to the right and 2 units up from your current location. How do we assign an n-tuple to a particular arrow?
5 Vectors If a displacement goes from initial location (a, b) to the terminal location (c, d). The move in the x 1 -direction is c a, since a + (c a) = a. The move in the x 2 -direction is d b, since b + (d b) = d. The displacement is (c a, d b) Generally, the displacement from the point p(a 1, a 2,..., a n ) to the point q(b 1, b 2,..., b n ) in R n is written pq = (b 1 a 1, b 2 a 2,..., b n a n )
6 The Algebra of Vectors Addition and subtraction: (3, 2) + (4, 1) = (7, 3) (x 1, x 2, x 3 ) + (y 1, y 2, y 3 ) = (x 1 + y 1, x 2 + y 2, x 3 + y 3 ) We can only add two vectors from the same vector space; vector addition is commutative u + v = v + u
7 The Algebra of Vectors Vector addition obeys the other rules which the addition of real numbers obeys: associative rule, the existence of a zero and the existence of an additive inverse. Zero vector represents no displacement at all; 0 = (0, 0,..., 0) Geometrically, it is a displacement PP having the same terminal point as initial point. If u = (a 1, a 2,..., a n ) the negative u is ( a 1, a 2,..., a n ); geometrically, one interchanges the head and the tail of u to obtain the head and tail of u ( PQ = QP)
8 Scalar Multiplication Go twice as far or you are halfway there We multiply a vector by a real number (or scalar) r x = (rx 1,..., rx n ) 2 (1, 1) = (2, 2) or 1 ( 4, 2) = ( 2, 1) 2
9 Scalar Multiplication Distributive laws in Euclidean spaces (r + s)u = ru + su r(u + v) = ru + rv where r and s are scalars and u and v are vectors.
10 Length and Inner Product in Rn The most basic geometric property is distance or length. Notation: the length of line segment PQ is denoted by the symbol PQ Consider P and Q lie in the plane R 2 and have the same x 2 -coordinate. P has (a 1, b) and Q has (a 2, b) PQ = a 2 a 1
11 Length
12 Length To find the distance from P(a 1, b 1, c 1 ) to Q(a 2, b 2, c 2 ) in R 3, we use PQ = (a 2 a 1 ) 2 + (b 2 b 1 ) 2 + (c 2 c 1 ) 2
13 Generalization If (x 1, x 2,..., x n ) and (y 1, y 2,..., y n ) are the coordinates of x and y, respectively, in Euclidean n-space, then x y = (x 1 y 1 ) 2 + (x 2 y 2 ) (x n y n ) 2 If we take y to be 0, then x = (x 1 ) 2 + (x 2 ) (x n ) 2
14 Scalar multiplication on the length Theorem rv = r v for all r in R 1 and v in R n Proof. r(v 1,..., v n ) = (rv 1,..., rv n ) = (rv 1 ) (rv n ) 2 = r 2 (v v n 2 ) = r (v v n 2 ), since r 2 = r
15 Unit Vector Example Given a non-zero displacement vector v, we will occasionally need to find a vector w which points in the same direction as v, but has length 1. Such a vector w is called the unit vector. To achieve such a vector w, premultiply v by the scalar r = 1 v For example the length of (1, 2, 3) in R 3 is (1, 2, 3) = ( 2) = (1, 2, 3) = (, , 3 14 ) It is a vector which points in the same direction as (1, 2, 3) but has length 1.
16 The Inner Product (dot product or scalar product) Definition Let u = (u 1,..., u n ) and v = (v 1,..., v n ) be two vectors in R n. The Euclidean inner product of u and v, written u v, is the number u v =u 1 v 1 + u 2 v u n v n Example if u = (4, 1, 2) and v = (6, 3, 4) then u v = = 13
17 The Inner Product Theorem Let u, v and w be arbitrary vectors in R n and let r be an arbitrary scalar. Then u v = v u u (v + w) = u v + u w u (rv) = r(u v) = (ru) v u u 0 u u = 0 implies u = 0 (u + v) (u + v) = u u+2(u v) + v v
18 The Inner Product The Euclidean inner product is closely connected to the Euclidean length of a vector u u = u1 2 + u un 2 and u = u1 2 + u u2 n u = u u Hence, the distance between two vectors u and v can be rewritten in terms of the inner product u v = (u v) (u v)
19 The Inner Product Theorem Let u and v be two vectors in the R n. Let θ the the angle between them. Then, u v = u v cos θ
20 The Inner Product Remarks: When the angle is a right angle, we say u and v are orthogonal u v = u 1 v u n v n = 0 Triangle Inequality. It states that any side of a triangle is shorter than the sum of the lengths of the other two sides. Theorem For any two vectors u and v in R n, u + v u + v
21 The Inner Product Theorem For any two vectors u and v in R n, Proof. Recall that u v u v u + v u + v = cos θ 1 then u v u v u (u v) + v 2 u u v + v 2 u u + u v + u v + v v ( u + v ) 2 (u + v) (u + v) ( u + v ) 2 u + v 2 ( u + v ) 2 u + v u + v
22 Line We will show how to describe lines and planes and their higher dimensional analogues. Let us work in R 2 y = mx + b m is the slope b is the y-intercept
23 Line What is the equation of line v in the figure? We cannot solve for y in terms of x
24 Parametric Representation A parametric representation of a point on a line uses parameter t in the coordinate expression of the point Expression (x 1 (t), x 2 (t)) for some value t of the parameter t Think of t as representing time and the parametrization as describing the transversal of a path. (x 1 (t), x 2 (t)) describes the particular location which is reached at time t.
25 Parametric Representation A line is determined by: a point x 0 on the line and a direction v in which to move from x 0. Geometrically, to describe motion in the direction v we add scalar multiples of v to x 0. x(t) = x 0 + tv
26 Parametric Representation Parameterization works in all dimensions. For example, the line in R 3 through the point x 0 = (2, 1, 3) in the direction v = (4, 2, 5) has the parameterization x(t) = (x 1 (t), x 2 (t), x 3 (t)) = (2, 1, 3) + t(4, 2, 5) = (2 + 4t, 1 2t, 3 + 5t)
27 Parametric Representation Another way to determine a line is to identify two points on the line. Suppose x and y lie on a line l. l can be viewed as the line which goes through x and points in the direction y x. A parameterization for the line is x(t) = x + t(y x) = x + ty tx = (1 t)x + ty We parameterize the line segment joining x to y as l(x, y) = {(1 t)x + ty :0 t 1}
28 Parametric Representation Let us consider two points x = (a, b) and y = (c, d) on line l in the plane. We can obtain the parameterized equation of l as x 2 b = d b c a (x 1 a)
29 Parametric Equation Let P be a plane on R 3. Let v and w be two vectors in P. For any scalar s and t, the vector sv + tw is called linear combination of v and w. A parameterization of the plane P is: x = sv + tw x 1 = sv 1 + tw 1 x 2 = sv 2 + tw 2 x 3 = sv 3 + tw 3
30 Parametric Equation If the plane does not pass through the origin but through point p =0 and if v and w are linearly independent vector from p: x = p + sv + tw
31 Parametric Equation To find the parametric equation of the plane containing the points, p, q and r, we have x(s, t) = p + s(q p) + t(r P) = (1 s t)p + sq + tr
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