Outside the high school program / Vectors 27 exercises corrected

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1. Non-orthogonal base E.487 The plane is equipped with an arbitrary coordinate system O ; −→ i ; −→ j , as shown below : 1 Determine the coordinates of points A , B , C , D , and E . 2 a Find the coordinates of vectors −−→ EC and −−→ BA . b Show that vectors −−→ EC and −−→ BD are collinear. 3 We use the function that takes two points in the plane and returns a real number, defined as follows : d ( A ; B ) = x B − x A 2 + y B − y A 2 4 a Determine the following values : d ( A;D ) ; d ( A;E ) b In an orthonormal coordinate system, the distance be-tween two points A and B is given by the formula : AB = x B − x A 2 + y B − y A 2 Does this formula make sense in any coordinate sys-tem? E.503 Add a coordinate system O ; I ; J to the diagram. 1 Plot the following points on the coordinate system shown here : A (3 ; 2) ; B ( − 3 ; 4) ; C ( − 1 ; − 5) 2 a Find the coordinates of the midpoint I of line seg-ment [ BC ] . b Find the coordinates of the midpoint J of line segment [ AB ] . c Plot point G , the center of mass of triangle ABC . 3 Suppose there exists a point H that satisfies the following statement : −→ OA + −−→ OB + −−→ OC = 3 · −−→ OH ( ∗ ) a Show that for any point M in the plane, we have : −−→ MA + −−→ MB + −−→ MC = 3 · −−→ MH b Conclude that : −−→ HA + −−→ HB + −−→ HC = −→ 0 . c Conclude the relation: −−→ AH = 2 3 · −→ AI . 4 a Prove that points G and H coincide. b Use the relationship ( ∗ ) to determine the coordinates of point G . https://chingmath.fr chapExoCorrec/487 sacados/487 −i−j-7-6-5-4-3-2-112345678-3-2-112ABCDE chapExoCorrec/503 sacados/503 −i−j−5−4−3−2−1234−5−4−3−2−1234567
E.513 In a plane with an arbitrary coordinate system O ; −→ i ; −→ j , consider the following three points de-fined by their coordinates : A (5 ; − 2) ; B ( − 3 ; − 1) ; C ( − 5 ; 3) 1 a Determine the coordinates of point M that satisfy the following relationship : 7 ·−−→ BM = 7 3 ·−−→ CM b Plot point M in the coordinate system. Verify graphi-cally that the three points B , C , and M are collinear. 2 a Determine the coordinates of point G that satisfy the vector relationship : −→ GA + −−→ GB + −−→ GC = −→ 0 b Place point G on the coordinate plane. c Draw the three medians of triangle ABC . What do you notice? E.497 Add an arbitrary coordinate system O ; −→ i ; −→ j to the plane, as shown below : 1 Draw a line segment representing each of the two vec-tors : −→ u 5 2 ; −→ v − 3 − 2 2 a Draw a line segment representing the vector −→ w de-fined by: −→ w = −→ u + −→ v b Graphically, determine the coordinates of vector −→ w . c Compare the coordinates of vector −→ w with those of vectors −→ u and −→ v . 2. Selected milestones E.514 We want to place point F on the figure below such that : F ∈ ( AB ) ; F , C , and E are aligned. We add the coordinate system A ; −−→ AD ; −−→ AB to the plane. 1 a Find the coordinates of the five points in the figure. b To which axis does point F belong? Find the x-coordinate F . Let f be the y-coordinate of point F . 2 a Determine the coordinates of vectors −−→ FC and −−→ FE . b Use this information to find the coordinates of point F . 3 Let I denote the midpoint of the segment [ DE ] . Show that the lines ( BI ) and ( FE ) are parallel. E.2896 In the plane, consider a parallelogram ABCD and the two points E and F defined by the relations : −→ AE = 5 3 · −−→ AD ; −→ AF = 5 2 · −−→ AB 1 Draw a representation of this configuration. 2 The plane is given the reference frame A ; −−→ AB ; −−→ AD . a Give, without justification, the coordinates of the points F , C and E . b Demonstrate that the points E , C and F are aligned. https://chingmath.fr chapExoCorrec/513 sacados/513 −i−j-5-4-3-2-112345-3-2-11234 chapExoCorrec/497 sacados/497 −i−j-1123456789-112345 chapExoCorrec/514 sacados/514 ABCDE chapExoCorrec/2896 sacados/2896
E.2108 Let ABCD be an unflattened paral-lelogram of center O . I is the midpoint of [ AB ] and J the midpoint of [ BC ] . The straight line ( DI ) intersects ( AC ) at M and the straight line ( DJ ) intersects ( AC ) at P . The plane is given the reference frame A ; −→ AI ; −→ AC : 1 Determine, by graphical reading, the coordinates of the points A , I , C and B 2 a Determine the two relative integers ¸ and ˛ achiev-ing equality: −−→ AD = ¸ · −−→ AB + ˛ · −→ AC b Deduce the coordinates of the point D . 3 a Show that the vector −→ BJ has coordinates : −→ BJ − 1 ; 1 2 b Deduce the coordinates of point J . 4 To which axis do the points M and P belong? Give the value of their abscissa. 5 Using the collinearity of the vectors −−→ DM and −→ DI , deter-mine the ordinate m of M . 6 Similarly, determine the ordinate p of P 7 a Determine the coordinates of the vectors −−→ AM , −−→ MP and −−→ PC . b Deduce that : AM = MP = PC . E.502 Consider a trapezoid ABCD satisfying the vector equality: −−→ DC = 1 3 −−→ AB . Let I and J be the midpoints of segments [ AB ] and [ CD ] , respectively. The lines ( AC ) and ( BD ) intersect at M , and the lines ( AD ) and ( BC ) intersect at N . 1 a Draw a diagram of this configuration. b Make a conjecture about the relative positions of points I , J , M , and N . Add an arbitrary coordinate system A ; −−→ AB ; −−→ AD to the plane : 2 Determine the coordinates of the points : A ; B ; D ; I ; C ; J 3 a To which axis does point N belong? Use this to find the x-coordinate of point N . b Let ¸ be the unique positive real number satisfying the equality: DN = ¸ · DA Using Thales’ theorem, determine the value of ¸ . c Use this to find the coordinates of point N . 4 a Prove that : AM = 3 4 · AC . b Use this to find the coordinates of point M . 5 a Determine the coordinates of vectors −−→ NJ , −→ NI , and −−→ NM . b Confirm the conjecture made in question 1 b . 3. Around the center of gravity of a triangle E.2078 Consider the plane provided with the reference frame O ; −→ i ; −→ j orthonormal : 1 Place the following points : https://chingmath.fr chapExoCorrec/2108 sacados/2108 chapExoCorrec/502 sacados/502 chapExoCorrec/2078 sacados/2078
A ( − 3 ; − 3) ; B (3 ; − 1) ; C ( − 1.5 ; 1) 2 a Determine the coordinates of point I midpoint of segment [ AB ] . b Determine the coordinates of point J midpoint of seg-ment [ AC ] . c Place on the figure the points I and J as well as the center of gravity G of the triangle ABC . 3 Consider the point M ( − 0.5 ; − 1) of the plane : a Calculate the coordinates of the vectors −→ CI and −−→ CM . b Show that the vectors −→ CI and −−→ CM . The collinearity coefficient of the vector −−→ CM will be determined as a function of the vector −→ CI . 4 Deduce the coordinates of the center of gravity of the triangle ABC . E.2895 1 In the plane, place three points A , B , C not aligned and the point I middle of the segment [ AB ] . 2 a Place the point M such that : −−→ AM =2 ·−−→ MC . b Place the point N such that : −−→ CN = −→ CA + −−→ CB 3 a Place the point G center of gravity of the triangle ABC . b Using the position of point G on median [ CI ] , estab-lish the following equality: −→ GA + −−→ GB + −−→ GC = −→ 0 4 The plane is given the reference frame A ; −−→ AB ; −→ AC any. a Give, without justification, the coordinates of the points A , B , C , I , M and N . b Using vector equality: −→ IC = −→ AC −−→ AI show that the point G has coordinates : G 1 3 ; 1 3 4. Algebraic manipulations E.505 Consider any parallelogram ABCD . We denote I , J , K , L the respective middles of the segments [ AD ] , [ AB ] , [ BC ] and [ CD ] . Establish the following two relationships : a −→ IJ + −→ IL = −−→ DC b 2 · −→ AJ + −−→ BD + −→ JB = −→ JC E.2056 1 Place two points A and B in the plane. 2 Consider the point M defined by the relation 2 · −−→ AM − 3 · −−→ MB = −→ 0 a Give an expression for the vector −−→ AM as a function of the vector −−→ AB . b Place the point M in the plane. E.2079 1 Consider four points A , B , C , D of the plane, verifying the following relationship : −→ AC − 2 · −−→ DA = 3 · −−→ BC Show that −−→ AB and −−→ CD are collinear. 2 Consider the parallelogram EFGH and I , J , K and L the respective middles of the sides [ EF ] , [ FG ] , [ GH ] and [ HE ] . Check the following relationship : −→ LG = −→ LI + 2 · −−→ HK + −→ IH E.526 1 a Place three points A , B , C not aligned in the plane. b Name I the middle of segment [ BC ] . 2 a Place the point G of the plane verifying the relation: 3 · −→ GI + −→ IA = −→ 0 . b Draw the sum representative : −→ GA + −−→ GB + −−→ GC 3 Algebraically, justify that the point G verifies the rela-tion : −→ GA + −−→ GB + −−→ GC = −→ 0 5. Selected landmark E.4591 Consider the square ABCD shown be-low : Its four sides have been divided into four equal parts. Con-sider the quadrilateral IJKL shown in the figure verifying: BI = CJ = DK = AL = 1 4 · AD Consider the plane provided with the reference frame A ; D ; B . 1 Give the coordinates of the eight points in this figure. 2 Demonstrate that the quadrilateral IJKL is a parallelo-gram. 3 Demonstrate that the parallelogram IJKL is a rectan-gle. 4 Demonstrate that the rectangle IJKL is a square. https://chingmath.fr chapExoCorrec/2895 sacados/2895 chapExoCorrec/505 sacados/505 chapExoCorrec/2056 sacados/2056 chapExoCorrec/2079 sacados/2079 chapExoCorrec/526 sacados/526 chapExoCorrec/4591 sacados/4591 ABCDIJKL
6. Reminders E.2439 Consider, in the plane, the two vectors −→ u and −→ v below : 1 Draw in the grid a representative −→ w of the sum −→ u + −→ v . 2 Draw in the grid a representative −→ y of the difference −→ u −−→ v . 3 Draw in the grid a representative −→ z of the following lin-ear combination : 2 −→ u +3 −→ v . E.2441 1 a Place the point D such that : −−→ CD = − 3 · −→ i +2 · −→ j . b Place the point F such that : −−→ EF = − 3 · −→ i − 4 · −→ j 2 Complete the following equality: −−→ GH = : : : · −→ i + : : : · −→ j 3 Complete the following dotted lines : a −→ u = : : : · −→ i b −→ v = : : : · −→ j c −−→ LM = : : : · −→ i + : : : · −→ j d −→ w = : : : · −→ i + : : : · −→ j e −→ z = : : : · −→ i + : : : · −→ j f −→ r = : : : · −→ i + : : : · −→ j g −→ s = : : : · −→ i + : : : · −→ j E.2442 1 Show that the vectors −→ u and −→ v are collinear by specify-ing the collinearity coefficient of −→ u and −→ v : a 1 2 · −→ u = 3 4 · −→ v b 3 · −→ u − 2 · −→ v = −→ 0 c 3 · −→ u − 2 −→ v = −→ 0 d − 2 · −→ u + −→ v = 2 · −→ u + 3 · −→ v 2 Which of the pairs of vectors below are collinear with each other? We will then specify the proportionality co-efficient of −→ u and −→ v : a −→ u (3 ; 2) et −→ v (9 ; 4) b −→ u (2 ; 3) et −→ v (4.2 ; 6.3) c −→ u ( − 1 ; 2) et −→ v (4 ; − 8) d −→ u (0.7 ; 4.1) et −→ v ( − 2.8 ; 16.4) 3 For two vectors −→ u and −→ v collinear, compare the collinearity coefficient of −→ v and −→ u relative to the collinearity coefficient of −→ u and −→ v . E.2440 Let ABCD be a parallelogram. Let I be the midpoint of [ AB ] and J the midpoint of [ DC ] . Determine a vector resulting from each of the expressions : a −−→ AB + −→ IJ − −→ DJ b −→ AC + −→ JA c −→ AI + −−→ AD E.2443 Let A , B , C be three points of the plane verifying the relation: − 1 2 · −−→ AB + 5 2 · −−→ BC − −−→ BA + −−→ CB = −→ 0 Show that the points A , B , C are aligned. https://chingmath.fr chapExoCorrec/2439 sacados/2439 chapExoCorrec/2441 sacados/2441 chapExoCorrec/2442 sacados/2442 chapExoCorrec/2440 sacados/2440 chapExoCorrec/2443 sacados/2443
E.2444 Consider the plane equipped with an orthogonal coordinate system : 1 Place the following points : A ( − 3 ; − 3) ; B (3 ; − 1) ; C ( − 1.5 ; 1) 2 a Determine the coordinates of point I , the midpoint of segment [ AB ] . b Determine the coordinates of point J , the midpoint of segment [ AC ] . c Place points I and J on the figure, as well as the center of gravity G of triangle ABC . d Determine the length IC . 3 Consider the point M ( − 0 ; 5 ; − 1) on the plane a Determine the coordinates of the vectors −−→ CM and −→ CI . b Show that points C , M , and I are collinear. c Using the collinearity coefficient between the two vec-tors −→ CI and −−→ CM , deduce that points M and G coin-cide. Hint: Remember that the center of gravity of a triangle is the point where the medians of the triangle intersect. The center of gravity of a triangle has the following metric property: ˇ The center of gravity is located on each median at 2 = 3 of the median starting from the vertex. ı E.2767 Consider the plane provided with a ( O ; I ; J ) orthonormal, the circle C of center K (2 ; 1) and ra-dius 2.5 , and the point A 0 ; 5 2 1 Show that the point A belongs to the circle C . 2 Determine the coordinates of the point B , belonging to the circle C , diametrically opposite the point A . 3 Let C 3 2 ; 1 − 6 , justify that the triangle ABC is right-angled at C . 4 Determine the coordinates of a point on the circle C , whose abscissa is 5 2 E.2445 Let ABCD be an unflattened paral-lelogram of center O . I is the midpoint of [ AB ] and J the midpoint of [ BC ] . The line ( DI ) intersects ( AC ) at M and the line ( DJ ) inter-sects ( AC ) at P . The aim of the problem is to show that : AM = MP = PC The ( A ; −→ AI ; −→ AC ) coordinate system will be used throughout the exercise: 1 Determine, by graphical reading, the coordinates of points A , I , C and B . 2 Deduce the coordinates of the points J and D (note that −−→ AD = −−→ BC ) 3 Determine the abscissa of points M and P . 4 Using the collinearity of the vectors −−→ DM and −→ DI , deter-mine the ordinate m of M . 5 Determine the ordinate p of P . 6 Compare the vectors −−→ AM , −−→ MP and −−→ PC . Deduce that : AM = MP = PC . 7. Vector and affine function E.970 In an orthonormal coordinate system O ; I ; J , the unit of measurement is the centimeter. 1 Plot the points A (2 ; 0) ; B (3 ; 5 ; 6) ; C (9 ; 5 ; 5) . 2 Plot point D in this coordinate system such that : −−→ AD = −−→ AB + −→ AC 3 Calculate the coordinates of vector −−→ BC . 4 Calculate the coordinates of the midpoint M of the seg- ment [ AC ] . 5 Let f be the linear function such that f (2)=0 and f (3 ; 5)=6 . Find the algebraic expression for f . 6 Plot the graph of f . https://chingmath.fr chapExoCorrec/2444 sacados/2444 -4-2024-4-224ij chapExoCorrec/2767 sacados/2767 chapExoCorrec/2445 sacados/2445 chapExoCorrec/970 sacados/970
E.971 Consider the plane with respect to the orthonormal coordinate system O ; I ; J : 1 Plot the points A ( − 7 ; 1) and B (1 ; 7) . 2 a What are the coordinates of the vectors −→ OA , −−→ OB , and −−→ AB ? Prove that AOB is an isosceles right triangle. b Let C be the circumcircle of triangle AOB . Calculate the coordinates of its center S and its radius. 3 Let f be the affine function defined by: f ( − 7) = 1 ; f (1) = 7 a Determine the algebraic expression for f . b In the coordinate system above, graph the function f . E.953 The plane is equipped with an or- thonormal coordinate system O ; I ; J . The unit of length is the centimeter. Consider the points : A (3 ; 1) ; B (2 ; − 2) ; C ( − 6 ; 4) Part 1 1 Plot the points A , B , and C in the coordinate system. 2 Consider the affine function f : x ↦→ mx + p , whose graph is the line ( AB ) . a Find the images of 2 and 3 under the function f . b Determine the values of m and p of the function f . Part 2 1 Show that : AC = 90 . 2 Given AB = 10 and BC =10 . Show that triangle ABC is a right triangle at A . 3 Calculate the coordinates of vector −−→ AB . 4 Construct point D , the image of point C under the trans-lation by vector −−→ AB . Determine the coordinates of point D graphically. 5 Show that quadrilateral ABDC is a rectangle. 6 Consider the circle C circumscribed around rectangle ABDC . Determine the coordinates of its center, then construct the circle C . 8. Unclassified financial years E.517 In a coordinate system O ; −→ i ; −→ j , consider the points : A (3 ; − 5) ; B ( − 2 ; 0) ; C (147 ; − 13) ; D ( − 53 ; 187) Show that the lines ( AB ) and ( CD ) are parallel. https://chingmath.fr chapExoCorrec/971 sacados/971 Clermont-Ferrand 2000 - 8 points -10-8-6-4-22I48JO chapExoCorrec/953 sacados/953 chapExoCorrec/517 sacados/517