Outside the high school program / Geometry in space: scalar product and plane 7 exercises (100% corrected)

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-3-2-12345I-12345JO -4-3-2-12345I-2-1234JO OIJABC 1. Introduction (via contact details) E.6647 In a plane with an orthonormal coordinate system O ; I ; J . Consider the points A , B and C defined by: A ( − 3 ; 1) ; B (4 ; − 1) ; C (1 ; 3) . 1 Determine the coordinates of the point I midpoint of the segment [ BC ] . 2 Let J be the image of point A by central symmetry of center C . a Give a vector relation verified by the points A , C and J . b Determine the coordinates of point J . 3 Determine the norm of the vector −−→ AB E.6646 A plane with an orthonormal O ; I ; J coordinate system is considered. 1 On Recalls the formula for the distance between two points : PQ = x Q − x P 2 + y Q − y P 2 Consider the three points in the plane A , B and C with coordinates : A ( − 3 ; 2) ; B ( − 2 ; − 2) ; C (2 ; − 1) a Déterminer the distances AB , AC and BC . b Establish that the triangle ABC is a right-angled tri-angle. 2 Soit −→ u ( x ; y ) a vector, we define the norm of the vector −→ u as the number −→ u defined by: −→ u = x 2 + y 2 Consider the two points E and F with coordinates : E ( − 1 ; 2) ; G (4 ; 3) and the two vectors −→ u and −→ v with coordinates : −→ u (4 ; − 1) ; −→ v (1 ; 2) a Déterminer the norms of the vectors −→ u and −→ v . b Déterminer the coordinates of the points F and H ver-ifying the two vector equalities: −−→ EF = −→ u ; −−→ HG = −→ v c Exprimer the vector −→ u + −→ v using the points E , F and G ? d Is the triangle EFG a rectangle? E.2572 Consider the plane provided with the orthonormal frame of reference ( O ; I ; J ) and three points A , B , C of the plane. We don’t know the coordinates of the points A and B but we note : −−→ AB ( x ; y ) ; −−→ BC ( x ; y ) 1 Determine the coordinates of the vector −→ AC . 2 Express the length of each of the vectors −−→ AB , −−→ BC , −→ AC as a function of x , x , y , y . They are respectively noted −−→ AB , −−→ BC , −→ AC . 3 Give a necessary and sufficient condition for the straight lines ( AB ) and ( BC ) to be perpendicular. 2. Scalar product and coordinates E.4091 In space provided with an orthonormal reference frame O ; −→ i ; −→ j ; −→ j , we give the points : A 2 ; 1 ; 3 ; B − 3 ; − 1 ; 7 ; C 3 ; 2 ; 4 1 Determine the coordinates of the point H barycenter of the system : https://chingmath.fr chapExoCorrec/6647 sacados/6647 -3-2-12345I-12345JO chapExoCorrec/6646 sacados/6646 -4-3-2-12345I-2-1234JO chapExoCorrec/2572 sacados/2572 OIJABC chapExoCorrec/4091 sacados/4091 Extrait de Liban Mai 2006
( A ; − 2) ; ( B ; − 1) ; ( C ; 2) 2 Determine the nature of the set Γ 1 , of points M of space such that : − 2 −−→ MA − −−→ MB + 2 −−→ MC · −−→ MB − −−→ MC = 0 Specify the characteristic elements. 3 Determine the nature of the set Γ 2 , of points M of space such that : − 2 −−→ MA − −−→ MB + 2 −−→ MC = √ 29 Specify the characteristic elements. 3. Set of points E.4104 Consider two points A and D of space and denote by I the midpoint of the segment [ AD ] . 1 Demonstrate that, for any point M of space : −−→ MD · −−→ MA = MI 2 − IA 2 2 Deduce the set ( E ) of points M of space, such that : −−→ MD · −−→ MA = 0 E.4107 Let A and B be two distinct points in the plane. 1 Characterize the set of points such that : a −−→ AB · −−→ AM = 0 b −−→ AB · −−→ AM = AB 2 c −−→ AB · −−→ AM = − AB 2 2 It is assumed that AB =6 . Characterize the set of points such that : a −−→ AB · −−→ AM = 18 b −−→ AB · −−→ AM = − 1 E.4109 In space, consider two distinct points A and B such that AB =4 . Let I be the midpoint of segment [ AB ] : 1 Demonstrate that for any point M of space, we have the equality: MA 2 − MB 2 = 2 · −−→ IM · −−→ AB 2 Determine the set of points M verifying: MA 2 − MB 2 = 16 https://chingmath.fr chapExoCorrec/4104 sacados/4104 chapExoCorrec/4107 sacados/4107 chapExoCorrec/4109 sacados/4109