Profesora: Eugenia Rosado. E.T.S. Arquitectura. Euclidean Geometry1 1 Euclidean vector space and euclidean a¢ ne space 1.1 Scalar product. Euclidean vector space. Let V be a real vector space. De…nition. A scalar product is a map (denoted by a dot ) V V R ! (~u;~v) ~u ~v 7! satisfying the following axioms: 1. commutativity ~u ~v = ~v ~u 2. distributive ~u (~v + ~w) = ~u ~v + ~u ~w 3. ( ~u) ~v = (~u ~v) 4. ~u ~u 0, for every ~u V 2 5. ~u ~u = 0 if and only if ~u = 0 De…nition. Let V be a real vector space and let be a scalar product. The pair (V; ) is said to be an euclidean vector space. Example. The map de…ned as follows V V R ! (~u;~v) ~u ~v = x1x2 + y1y2 + z1z2 7! where ~u = (x1; y1; z1), ~v = (x2; y2; z2) is a scalar product as it satis…es the …ve properties of a scalar product. This scalar product is called standard (or canonical) scalar product. The pair (V; ) where is the standard scalar product is called the standard euclidean space. 1.1.1 Norm associated to a scalar product. Let (V; ) be a real euclidean vector space. De…nition. A norm associated to the scalar product is a map de…ned as follows V kk R ! ~u ~u = p~u ~u: 7! k k Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.2 1.1.2 Unitary and orthogonal vectors. Orthonormal basis. Let (V; ) be a real euclidean vector space. De…nition. A vector ~u V is said to be unitary if ~u = 1. 2 k k Given a vector ~u V , ~u = ~0, we can obtain a unitary vector ~v from ~u just considering the vector 2 6 1 ~v = ~u ~u k k as 1 1 ~v = ~u ~u = ~u ~u = 1: k k k k k k k k De…nition. Two vectors ~u;~v V are said to be orthogonal is if ~u ~v = 0. 2 Theorem. Let ~u;~v; ~w be three nonzero vectors in a 3-dimensional eucliden vector space (V; ). If the vectors ~u;~v; ~w are two by two orthogonal then B = (~u;~v; ~w) is a basis of (V; ). (The proof is left to the reader). De…nition. Let (V; ) be a real euclidean vector space. A basis B of (V; ) is said to be orthonormal if the vectors in B are unitary and two by two orthogonal. Example. Let (V; ) be the standard euclidean space and let ~u = (1; 0; 1), ~v = ( 1; 0; 1), ~w = (0; 3; 0). Let check that B = (~u;~v; ~w) is a basis of (V; ) and let obtain an orthogonal basis of (V; ). The vectors ~u;~v; ~w are two by two orthogonal: ~u ~v = (1; 0; 1) ( 1; 0; 1) = 0; ~u ~w = (1; 0; 1) (0; 3; 0) = 0; ~v ~w = ( 1; 0; 1) (0; 3; 0) = 0; hence B = (~u;~v; ~w) is a basis of V . As ~u = p2, ~v = p2 and ~w = 3, B is not an orthonormal basis. By considering the vectorsk k k k k k 1 1 1 ~u0 = ~u = ; 0; ; ~u p2 p2 k k 1 1 1 ~v0 = ~v = ; 0; ; ~v p2 p2 k k 1 ~w0 = ~w ~w = (0; 1; 0); k k we obtain an orthonormal basis B0 = (~u0;~v0; ~w0). Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.3 1.1.3 Angle between two vectors. De…nition. Let ~u;~v be nonzero vectors in an eucliden vector space (V; ). The cosine of the angle given by the two vectors ~u;~v is de…ned as the real number satisfying ~u ~v cos(~u;~v) = ; ~u ~v k k k k d where ~u;~v denotes the angle given by the two vectors ~u;~v. d 1.1.4 Vector product. Let (V; ) be a 3-dimensional euclidean vector space with orthonormal basis B = (~u1; ~u2; ~u3). De…nition. We de…ne the vector product of two vectors ~u = (x1; y1; z1), ~v = (x2; y2; z2), and we denote it ~u ~v, as the vector with components ^ y z x z x y 1 1 ; 1 1 ; 1 1 y2 z2 x2 z2 y2 y2 respect to the basis B. Simbolically ~u ~v can be writen as folows ^ ~u1 ~u2 ~u3 ~u ~v = x1 y1 z1 ^ x2 y2 z2 y1 z1 x1 z1 x1 y1 = ~u1 ~u2 + ~u3: y2 z2 x2 z2 y2 y2 Remark. The norm of the vector product of two vector ~u, ~v satis…es ~u ~v = ~u ~v sin ~u;~v : k ^ k k k k k Properties. The most important propeties of the vectord product are: 1. Anticommutativity ~u ~v = ~v ~u ^ ^ 2. ( ~u) ~v = (~u ~v) ^ ^ 3. ~u (~v + ~w) = ~u ~v + ~u ~w ^ ^ ^ 4. The vector product ~u ~v is orthogonal to ~u and to ~v. ^ 5. The vector product of two vector ~u, ~v is zero if and only if the two vectors are linearly dependet. (The proof is left to the reader). Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.4 1.1.5 Mixed product. Let (V; ) be a 3-dimensional euclidean vector space with orthonormal basis B = (~u1; ~u2; ~u3). De…nition. The map V V V R ! (~u;~v; ~w) ~u (~v ~w) ; 7! ^ is called the mixed product. Remark. If ~u = (x1; y1; z1), ~v = (x2; y2; z2) and ~w = (x3; y3; z3) and taking into account that B = (~u1; ~u2; ~u3) is an orthonormal basis, we have y2 z2 x2 z2 x2 y2 ~u (~v ~w) = (x1~u1 + y1~u2 + z1~u3) ~u1 ~u2 + ~u3 ^ y3 z3 x3 z3 y3 y3 y2 z2 x2 z 2 x2 y 2 = x1 y1 + z1 y3 z3 x3 z3 y3 y3 x 1 y1 z1 = x y z : 2 2 2 x3 y3 z3 Remark. Geometrically, the absolute value of the mixed product represents the volume of the parallelepiped whose edges concur in the same vertex are the three vectors. 1.2 Euclidean a¢ ne space De…nition. An a¢ ne space (A; V; ) is said to be euclidean if the associated vector space V is an euclidean vector space. Usually an euclidean a¢ ne space is denoted E. De…nition. In an euclidean a¢ ne space E we de…ne the distance of two ponts as follows: d(P; Q) = !PQ We will cosider orthonormal a¢ ne references; that is, references = O; B = (~u1; ; ~un) such that the basis B is an orthonormal basis. R f g Property. The change of coordinates matrix of two orthonormal references , 0, R R 1 0:::0 b1 M( 0; ) = 0 . 1 R R . A B C B bn C B C t 1 t @ A satis…es A = A (where A denotes the transpose matrix of A). That is, the matrix A is orthogonal. Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.5 1.3 Metric problems. Let E be an euclidean a¢ ne space. De…nition. A vector ~v V is said to be orthogonal to a¢ ne subspace L E if ~v is orthogonal to the vector in2L~ . 1.3.1 Normal plane equation. Let A be a point in an a¢ ne plane and let ~v V be orthogonal to . Hence, a point X is in the plane if and only if 2 !AX ~v = 0: A vector ~u satisfaying !AB ~u = 0 for all A; B is said to be a normal vector to the plane. 2 If ~v V be orthogonal to is a unitary vector, then !AX ~v = 0 is called the normal plane equation2 . 1.3.2 Angle between two lines. Let r1 be the line containing a point A and with director vector ~u, and let r2 be the line containing a point B and with director vector ~v. The angle between the lines r1 and r2 is the angle between the vectors ~u and ~v. 1.3.3 Angle between two planes. Let 1 be the plane containing a point A and with normal vector ~u, and let 2 be the plane containing a point B and with normal vector ~v. The angle between the planes 1 and 2 is the angle between the normal vectors ~u and ~v. 1.3.4 Angle between a line and a plane. Let r be the line containing a point A and with director vector ~u, and let be the plane containing a point B and with normal vector ~v. The angle between the line r and the plane is the complementary of the angle between the vectors ~u and ~v; that is, sin = cos( ) = cos(~u;~v): 2 d Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.6 1.3.5 Distance between a point and a line. The distance between a point P0 and a line r is the minimum of the distances between P0 and P where P r. Let P r and2 let ~v be a director vector of r, hence 2 \ !P0P ~v = !P0P ~v sin(!P0P ;~v) ^ k k = ~v d( P0; r); k k and, therefore !P0P ~v d(P ; r) = ^ : 0 ~v k k Example. Let compute the distance between the point A = (2; 3; 0) and the line r x 1 y z+2 1 = 1 = 2 . We have P = (1; 0; 2) r and ~v = ( 1; 1; 2) ~r hence 2 2 !AP = ( 1; 3; 2) ~u1 ~u2 ~u3 !AP ~v = 1 3 2 = (8; 4; 2) ^ 1 1 2 ~v = p1 + 1 + 4 = p 6; k k therefore !AP ~v p82 + 42 + 22 p4222 + 2222 + 22 d(A; r) = ^ = = ~v p6 p6 k k 22 (42 + 22 + 1) 2p21 = = : p p6 p6 Profesora: Eugenia Rosado, E.T.S.