Grade 12 / Vectors, lines and planes in space 35 exercises (100% corrected)

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ABCDEFGHIJKL ijkOABCDEFGH ABCDEFGHIJKLMNOPQRSTUVWXYZ 1. Space vectors E.2759 In space, consider the parallelepiped ABCDEFGH . Let I , J , K , L be the respective middles of the edges [ AB ] , [ CD ] , [ EF ] , [ GH ] . 1 a Give all vectors equal to the vector −→ AJ . b Give all vectors equal to vector −−→ ED . c Give all vectors equal to vector −−→ DK . 2 Give a representative of each of the following sums : a −−→ AD + −→ LF = −→ : : : b −→ DI + −−→ BF + −→ HI = 2 · −→ : : : c −−→ HB + −−→ CD = −−−→ : : : B E.2771 Space is provided with a O ; −→ i ; −→ j ; −→ k ; this reference frame and the associated grid is shown below : Determine the coordinates of the points A , B , C , D , E , F , G , H . E.2760 The cube shown below is made up of 8 identical cubes placed side by side : 1 In each question, determine the vector, having origin A representing the sum : a −−→ MY + −−→ KW b −→ A Ω + −→ V S + −−→ CX c −−→ Ω H + −→ ZE + −−→ QC d −→ AG + −−→ BH + −→ OI 2 In each question, determine the vector, having extremity A , representing the sum : a −→ U Ω + −→ V L b −→ LZ + −−→ XZ + −−→ SX c −−→ Y Q + −→ AY + −→ SA d −−→ X Ω + −−→ XH + −→ GZ https://chingmath.fr chapExoCorrec/2759 sacados/2759 ABCDEFGHIJKL chapExoCorrec/2771 sacados/2771 ijkOABCDEFGH chapExoCorrec/2760 sacados/2760 ABCDEFGHIJKLMNOPQRSTUVWXYZ
BDFHIJKLMNOPQRSTUVWXY−i−j−k ABCDEFGH ABCDEFGHIJKL E.4180 In the space O ; −→ i ; −→ j ; −→ k , consider the cubes below : 1 Which points on the figure verify y 0 ? 2 a Is the right paving stone PQJKXY ST included in the half-space defined by the inequation : x 0 b Is the right paving stone FDBHWUSY included in the half-space defined by the inequation : z 0 3 Describe the following half-spaces : a x 0 b z 0 c y − z 0 2. Relative position E.6816 Consider the cube ABCDEFGH shown below : 1 Give the relative position of the following pairs of straight lines : a ( EH ) and ( BC ) b ( EB ) and ( FA ) c ( BA ) and ( EG ) d ( EC ) and ( AG ) 2 Give the relative position of the following pairs of lines and planes : a ( EH ) and ( AFG ) b ( HD ) and ( FAG ) c ( FA ) and ( DHG ) d ( BC ) and ( HFA ) 3 Give the relative position of the following planes : a ( HED ) and ( BCF ) b ( HGA ) and ( DCB ) E.6817 In space let’s consider the cube ABCDEFGH and the points I , J , K , L respective middles of the segments [ EH ] , [ CG ] , [ AB ] , [ AD ] . Give the relative positions of the following pairs of straight lines : a ( IL ) and ( FB ) b ( AG ) and ( KJ ) c ( KL ) and ( HF ) d ( EL ) and ( HD ) https://chingmath.fr chapExoCorrec/4180 sacados/4180 BDFHIJKLMNOPQRSTUVWXY−i−j−k chapExoCorrec/6816 sacados/6816 ABCDEFGH chapExoCorrec/6817 sacados/6817 ABCDEFGHIJKL
ABCDEFGH MNABCDEFGH ABCDEFGHKLMPQ E.2885 Consider the ABCDEFGH paral-lelepiped shown below : Let O be the midpoint of segment [ AG ] : 1 Show that point O is the midpoint of segment [ EC ] . 2 Demonstrate that point O is midpoint of segment [ HB ] . E.2775 Consider the cube ABCDEFGH and the points M and N of space. 1 Suppose the point M belongs to the plane ( EFB ) ; jus-tify that the straight lines ( AD ) and ( HM ) are non-coplanar. 2 Suppose the point M belongs to the plane ( EHD ) : a Justify that the straight lines ( AD ) and ( HM ) are coplanar. b Place the point L intersection of the straight lines ( AD ) and ( HM ) . 3 Depending on the position of the point N in space, spec-ify whether the straight lines ( GF ) and ( HN ) are copla-nar ; if so, place their point of intersection : a N ∈ ( HEF ) b N ∈ ( HDC ) E.2795 Consider the cube ABCDEFGH . The points K , L , M are the respective middles of the edges [ EH ] , [ HG ] , [ EF ] . The points P and Q represent the intersection of the straight line ( KL ) with the planes ( ABF ) and ( CBF ) respectively. 1 a Justify that the point P is the intersection of the straight lines ( EF ) and ( KL ) . b Deduce that point K is the midpoint of segment [ PL ] . We admit that by similar reasoning, the point L is the middle of the segment [ KQ ] : 2 Justify the following vector equality: −−→ PQ = 3 2 · −→ AC https://chingmath.fr chapExoCorrec/2885 sacados/2885 ABCDEFGH chapExoCorrec/2775 sacados/2775 MNABCDEFGH chapExoCorrec/2795 sacados/2795 ABCDEFGHKLMPQ
ABCDEFGH ABCDEIJ E.585 Definition: Two lines are coplanar if they belong to the same plane. In the cube ABCDEFGH shown opposite : 1 Which of the pairs of lines below are coplanar with each other? a ( EA ) et ( FB ) b ( HE ) et ( CB ) c ( HC ) et ( AD ) d ( GA ) et ( CA ) e ( HB ) et ( DA ) In the following question, we will use the following three def-initions : Definitions: Two lines are parallel in space if they are coplanar and if they are parallel in that plane. Two planes are parallel when they have no points in common or when they coincide. A line and a plane are parallel when : either P and Δ have no points in common. or the line Δ is included in the plane P 2 Which of the following pairs define a pair of parallel ob-jects : a ( GD ) et ( AB ) b ( EB ) et ( HGC ) c ( EF ) et ( DC ) d ( BAH ) et ( GFH ) Vocabulary : We only talk about perpendicular lines in the case of coplanar lines. Definition: Two lines are orthogonal if they are respectively paral-lel to two perpendicular lines in the same plane. A line is orthogonal to a plane if it is orthogonal to all lines in that plane. 3 Give the pairs below that are orthogonal : a ( EF ) et ( HE ) b ( DB ) et ( AB ) c ( HD ) et ( ABC ) d ( HB ) et ( BFG ) e ( AC ) et ( HDF ) f ( HF ) et ( GCF ) 3. Parallelism E.2766 The figure below shows the ABCDE pyramid with a square base ; the points I and J represent the respective middles of the [ BE ] and [ CE ] edges. 1 Justify that points A , D , I , J are coplanar. 2 a Justify that the straight lines ( AI ) and ( DJ ) are secant. b Note M their point of intersection. Place the point M in the figure above. 3 Deduce the line of intersection of planes ( ABE ) and ( CDE ) . 4. Locating in space E.2814 In space provided with the refer-ence frame A ; −→ i ; −→ j ; −→ k , consider the rectangular paral-lelepiped ABCDEFGH and the two points M and N . https://chingmath.fr chapExoCorrec/585 sacados/585 ABCDEFGH chapExoCorrec/2766 sacados/2766 ABCDEIJ chapExoCorrec/2814 sacados/2814
ijkABCDEFGHMN xyzMMNNPP xyzMMNNPP ABCDEFGHO 1 Determine the coordinates of point M when : a M is a point of the plane ( EFB ) . b M is a point of the plane ( HGD ) . 2 Determine the coordinates of point N when : a N is a point of the plane ( EFH ) . b N is a point of the plane ( EAD ) . E.2793 Consider the plane provided with a refer-ence frame O ; −→ i ; −→ j ; −→ k orthonormal represented below : Consider the three points M , N , P : 1 a The point M is the orthogonal project of M on the plane ( Oxy ) . Determine the coordinates of the point M in this plane. b The point N is the orthogonal project of N onto the plane ( Oyz ) . Determine the coordinates of the point N in this plane. c The point P is the orthogonal project of P onto the plane ( Oxz ) . Determine the coordinates of the point P in this plane. 2 Determine the coordinates of the three points M , N , P in this frame of reference. E.2776 We provide an orthonormal frame of reference whose graduations on the axes are shown ; consider the three points M , N , P and their respective projections M , N , P onto the plane ( OIJ ) , whose representations are as follows : 1 a Determine the coordinates of the point M belong-ing to the plane ( OIJ ) . b Give the coordinates of the point M . 2 Determine the coordinates of point N and point P . E.4024 Consider the ABCDEFGH cube shown below with center O . Determine the coordinates of the points of each of the vertices of the cube as well as of the point O in each of the following landmarks : a B ; −−→ BC ; −−→ BA ; −−→ BF b A ; −→ AE ; −−→ AB ; −−→ AD c O ; −−→ OF ; −−→ OG ; −−→ OE 5. Colinearity E.2779 1 Show that the following pairs of vectors are collinear: a −→ u 6 ; 21 ; 9 ; −→ v 4 ; 14 ; 6 b −→ u 3 ; 5 ; 4 3 ; −→ v 6 5 ; 2 ; 8 15 https://chingmath.fr ijkABCDEFGHMN chapExoCorrec/2793 sacados/2793 xyzMMNNPP chapExoCorrec/2776 sacados/2776 xyzMMNNPP chapExoCorrec/4024 sacados/4024 ABCDEFGHO chapExoCorrec/2779 sacados/2779
ABCDEFGHIKJ −nOABCVue en perspective cavalière de la réπexiond’un rayon lumineux sur le plan(OAB ABCDOIS 2 Justify that the following two vectors are not collinear: −→ u 5 ; 8 ; 3 ; −→ v 3 ; 24 5 ; 8 5 E.2792 Consider the ABCDEFGH cube shown opposite and the three points defined by: The point K is the mid-dle of [ EF ] ; the point I verifies the re-lation −→ AI = 1 3 ·−−→ AB ; the point J verifies the re-lation −→ AJ = 2 3 ·−−→ AD . Using the benchmark A ; −−→ AB ; −−→ AD ; −→ AE , show that the straight lines ( IJ ) and ( KH ) are parallel. E.6880 A retro-reflector is an optical de-vice formed by three mirrors in the shape of a ˇ cube ı corner, with the reflecting faces facing inwards. They are found in the reflectors of some vehicles, as well as in surveying equipment. The points O , A , B and C are vertices of a cube, so that the reference frame O ; −→ OA ; −−→ OB ; −−→ OC is an orthonormal coordi-nate system. This coordinate system will be used throughout the exercise. The three mirrors of the retro-reflector are represented by the planes ( OAB ) , ( OBC ) and ( OAC ) . Light rays are modeled by straight lines. Rules for reflecting a light ray admitted : when a light ray with director vector −→ v a ; b ; c is re-flected by the plane ( OAB ) , a director vector of the re-flected ray is −→ v a ; b ; − c ; when a light ray with director vector −→ v a ; b ; c is re-flected by the plane ( OBC ) , a direction vector of the reflected ray is −→ v − a ; b ; c ; when a light ray of direction vector −→ v a ; b ; c is reflected by the plane ( OAC ) , a direction vector of the reflected ray is −→ v a ; − b ; c ; Reflector property Using the previous rules, demonstrate that if a light ray of direction vector −→ v a ; b ; c is reflected successively by the planes ( OAB ) , ( OBC ) and ( OAC ) , the final radius is paral-lel to the initial radius. E.6886 Consider the regular pyramid SABCD with vertex S consisting of the square base ABCD and equilateral triangles shown below. Point O is the center of base ABCD with OB =1 . Recall that the segment [ SO ] is the height of the pyramid and that all edges are of equal length. 1 Justify that the coordinate system O ; −−→ OB ; −−→ OC ; −→ OS is orthonormal. In the rest of the exercise, we will use the coordinate system O ; −−→ OB ; −−→ OC ; −→ OS . 2 Define point K by the relation −−→ SK = 1 3 · −→ SD and denote I as the midpoint of segment [ SO ] . a Determine the coordinates of point K . b Deduce that points B , I , and K are collinear. c Let L be the point of intersection of edge [ SA ] with plane ( BCI ) . Justify that lines ( AD ) and ( KL ) are parallel. d Determine the coordinates of point L . 6. Space objects https://chingmath.fr chapExoCorrec/2792 sacados/2792 ABCDEFGHIKJ chapExoCorrec/6880 sacados/6880 −nOABCVue en perspective cavalière de la réπexiond’un rayon lumineux sur le plan(OAB chapExoCorrec/6886 sacados/6886 Extrait d'Antilles-Guyane Juin 2016 ABCDOIS
(d(dAB (d(dAB ABCDEFGHIJ PABCABC P(d(dABAB PABCDEFGHIJLK E.579 Verify that on the two drawings below, A and B are the points of intersection of intersection respec-tively of the straight lines ( d ) and ( d ) with the plane ( P ) E.588 Answer yes or no to the following ques-tions : 1 Do two points always define a single straight line? 2 Three points always define a single plane? 3 The intersection of two planes is a point? E.581 In space, consider the cube ABCDEFGH whose representation is given below : In the following representation, I is a point belonging to the line ( GH ) and J belongs to the line ( AB ) . Four statements are proposed below. Say whether each of these is true or false, justifying your answer. 1 triangle EHD rectangle H . 2 The straight lines ( AC ) and ( GH ) intersect at I . 3 The quadrilateral BCHE is a rectangle. 4 J is the point of intersection of ( CG ) and ( AB ) . E.587 Consider in space a plane ( P ) and A , B , C not belonging to this plane and not aligned. We note : A the point of intersection of the line ( BC ) with ( P ) ; B the point of intersection of the line ( AC ) with ( P ) ; C is the point of intersection of the line ( AB ) with ( P ) ; Show that the points A , B and C are aligned. E.584 In space, consider a plane ( P ) and two parallel straight lines ( d ) and ( d ) . A and B are respectively points on the straight lines ( d ) and ( d ) . We name A and B the respective points of intersection of the straight lines ( d ) and ( d ) with the plane ( P ) . 1 Justify that the points A , B , A and B are coplanar. 2 Justify that, on the graph, the straight lines ( AB ) and ( A B ) are not parallel. 3 Place, in the figure, the point of intersection of the lines ( AB ) and ( A B ) . 7. Impact rules E.2768 ABCDEDFGH is a cube ; consider the plane ( P ) whose section with the cube is the quadrilateral IJKL https://chingmath.fr chapExoCorrec/579 sacados/579 (d(dAB (d(dAB chapExoCorrec/588 sacados/588 chapExoCorrec/581 sacados/581 ABCDEFGHIJ chapExoCorrec/587 sacados/587 PABCABC chapExoCorrec/584 sacados/584 P(d(dABAB chapExoCorrec/2768 sacados/2768 PABCDEFGHIJLK
ABCDEFGHIJKLMN ABCDE ABCDEFGHIJKL Justify that IJKL is a parallelogram. E.6885 ABCDEFGH is a cube with edges equal to 1 . Let I be the midpoint of [ AB ] . Let P be the plane parallel to the plane ( BGE ) and passing through the point I . We assume that the section of the cube by the plane P shown above is a hexagon whose vertices I , J , K , L , M , and N be-long respectively to the edges [ AB ] , [ BC ] , [ CG ] , [ GH ] , [ HE ] , and [ AE ] , respectively. Show that point N is the midpoint of segment [ AE ] . 8. Roof theorem E.5404 Consider the square-based pyramid ABCDE , shown below : 1 Determine the position of the line ( d ) intersection of the planes ( ABE ) and ( CDE ) . 2 Represent the straight line ( d ) . E.2796 Consider the cube ABCDEFGH . The points I , J , K , L are the respective midpoints of the edges [ AD ] , [ DC ] , [ EH ] , [ HG ] . The point Q is the intersection point of the line ( KL ) and the plane ( CBF ) . Point S is the intersection point of line ( IJ ) and plane ( CBF ) . 1 Place points Q and S on the figure. 2 Prove that lines ( QB ) and ( FS ) are coplanar and inter-secting. Note M as their point of intersection. 3 Deduce the line ( d ) , the intersection of planes ( KLB ) and ( FIJ ) . 9. Unclassified financial years https://chingmath.fr chapExoCorrec/6885 sacados/6885 ABCDEFGHIJKLMN chapExoCorrec/5404 sacados/5404 ABCDE chapExoCorrec/2796 sacados/2796 ABCDEFGHIJKL
ABCDEFGHIJ OIJABCDE IJKPP ABCDEFGHIJKL E.6818 Consider the cube ABCDEFGH shown below : Let I and J be the respective middles of segments [ AE ] and [ BC ] . Of the four statements below, only one is correct. Indicate which one and justify your choice. a The straight lines ( IJ ) and ( EC ) are strictly parallel. b The straight lines ( IJ ) and ( EC ) are non-coplanar. c The straight lines ( IJ ) and ( EC ) are secant. d The straight lines ( IJ ) and ( EC ) are coincident. E.4913 Consider the pyramid ABCDE with rectangular base ABCD shown below : Note I and J the respective middles of segments [ EB ] and [ EC ] , and O the center of rectangle ABCD . 1 a Justify that the straight lines ( IJ ) and ( BC ) are parallel. b Justify that the straight lines ( IJ ) and ( AD ) are par-allel. 2 Justify that the straight line ( AD ) is parallel to the plane ( OIJ ) . E.4948 Let ( P ) and ( P ) be two secant planes. Let I , J be two points of the ( P ) plane and K a point of the ( P ) plane. Draw the line of intersection of the planes ( P ) and ( IJK ) . Justify your approach. E.4942 Consider the cube below où the points I , J , K and L are the respective middles of the segments [ AB ] , [ AD ] , [ HG ] and [ GF ] . 1 Justify that the straight lines ( IJ ) and ( BD ) are parallel. 2 a Justify that points H , D , B and F are coplanar. b Justify that the straight lines ( HF ) and ( DB ) are parallel. c Demonstrate that points I , J , K , L are coplanar. 3 What is the nature of the quadrilateral IJKL ? https://chingmath.fr chapExoCorrec/6818 sacados/6818 Extrait d'Antilles-Guyanne Juin 2013 ABCDEFGHIJ chapExoCorrec/4913 sacados/4913 OIJABCDE chapExoCorrec/4948 sacados/4948 IJKPP chapExoCorrec/4942 sacados/4942 ABCDEFGHIJKL
ABCDEFGHIJKL ABCDIJE (d1(d2P1P2 E.580 In space, consider the parallelepiped ABCDEFGH . Let I and J be the respective middles of the edges [ EF ] and [ FG ] . Note K the intersection of the straight lines ( AE ) and ( BI ) ; note L the point of intersection of the straight lines ( BJ ) and ( CG ) . 1 Justify that the points I , J , K and L belong to the same plane. 2 Show that the straight lines ( IJ ) and ( KL ) are parallel. E.583 In space, consider the pyramid ABCDE with a square base ; let I and J be the midpoints of the edges [ EB ] and [ EC ] , respectively: 1 a Prove that the points A , D , I , and J lie in the same plane. b Prove, without using any graphical arguments, that the lines ( AI ) and ( DJ ) intersect. c Let M denote the point of intersection of the lines ( AI ) and ( DJ ) . Plot the point M on the graph. 2 a Prove that the point M lies in the planes ( AEB ) and ( DEC ) . b Conclude that the line ( EM ) is the line of intersection of the planes ( AEB ) and ( DEC ) . 3 Conclude that ( EM ) is parallel to the base ABCD of the pyramid. E.5503 In space, consider two planes ( P 1 ) and ( P 2 ) respectively containing the lines ( d 1 ) and ( d 2 ) parallel to each other. If the planes ( P 1 ) and ( P 2 ) are secant then the straight lines ( d 1 ) and ( d 2 ) are parallel to the line of intersection of these two planes. https://chingmath.fr chapExoCorrec/580 sacados/580 ABCDEFGHIJKL chapExoCorrec/583 sacados/583 ABCDIJE chapExoCorrec/5503 sacados/5503 (d1(d2P1P2