Outside the high school program / Scalar product in space 3 exercises corrected

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1. Median formula E.6807 Reminders: Let A and B be two points on the plane and I the midpoint of the segment [ AB ] . For any point M on the plane, we have the relationship : AM 2 + BM 2 = 1 2 · AB 2 + 2 · MI 2 Consider the plane equipped with an orthonormal coordinate system O ; I ; J and the three points A , B , and C with re-spective coordinates : A ( − 2 ; − 3) ; B ( − 1 ; 2) ; C (3 ; 1) 1 Determine the measures AB , AC , and BC . 2 a Let I be the midpoint of segment [ AB ] . Determine the measure of the median in triangle ABC originating at vertex C . b Let J be the midpoint of segment [ AC ] . Determine the length of the median in triangle ABC originating at vertex B . E.2663 Consider a triangle ABC right-angled at A with the following measures : AB =6 ; AC = 3 Let I be the midpoint of segment [ AB ] and J the midpoint of [ IC ] . We are interested in the set E of points M verifying the rela-tion : MA 2 + MB 2 +2 · MC 2 =72 1 er method : 1 Show that all points M verify the relation: MA 2 + MB 2 + 2 × MC 2 = 4 · MJ 2 + JA 2 + JB 2 + 2 × JC 2 2 Using the median theorem twice, demonstrate the follow-ing relationship : M ∈E ⇐⇒ 4 × MJ 2 + AB 2 2 + IC 2 =72 3 Deduce the nature of the set E . 2 ème method : The plane is given the reference frame A ; 1 6 · −−→ AB ; 1 3 · −→ AC 1 Determine the coordinates of points A , B , C in this frame. 2 Noting ( x ; y ) the coordinates of point M , determine an equation of E in this reference frame. 3 Deduce the nature of the set E . E.2609 Consider the plane provided with a reference frame ( O ; I ; J ) orthonormal and the following two points in the plane : A ( − 3 ; 2) ; B (3 ; − 6) 1 We will denote by M the point with coordinates ( x ; y ) : a Determine the coordinates of point I middle of [ AB ] . b Determine the coordinates of the vectors −−→ IM and −−→ AB . c Determine the length AB . 2 a Determine the Cartesian equation of the set of points verifying the relation: MA 2 − MB 2 = 40 b Give the nature and characteristic elements of the points M verifying this relation. 3 a Determine the equation of the set of points verifying the relation: −−→ MA · −−→ MB = 34 b Give the nature and characteristic elements of the points M verifying this relationship. 4 a Determine the equation of the set of points verifying the relation: MA 2 + MB 2 = 150 b Give the nature and characteristic elements of the points M verifying this relation. https://chingmath.fr chapExoCorrec/6807 sacados/6807 chapExoCorrec/2663 sacados/2663 chapExoCorrec/2609 sacados/2609