Grade 11 / Scalar product 94 exercises (including 87 corrected)

a
-8-7-6-5-4-3-2-1234567I-4-3-2-12345JOA1B1A2B2A3B3A4B4 IJO ChingQuizz : 5 exercises available for Quizz assessment : 1. Reminders E.6486 Definition: in the plane equipped with an orthonormal co-ordinate system O ; I ; J , i.e., A ( x A ; y A ) and B ( x B ; y B ) : The coordinates of −−→ AB are: −−→ AB ( x B − x A ; y B − y A ) In the orthonormal coordinate system ( O ; I ; J ) below, four vectors are represented : Graphically, determine the coordinates of these four vectors. E.6481 Proposition: In the plane equipped with an orthonormal coordinate system O ; I ; J , let A ( x A ; y A ) and B ( x B ; y B ) be two points. The distance AB is defined by: AB = x B − x A 2 + y B − y A 2 Let K be the midpoint of the segment [ AB ] . The coor-dinates of the point K are: K x A + x B 2 ; y A + y B 2 In the plane equipped with an orthonormal coordinate system O ; I ; J , consider the following three points : A ( − 4 ; − 2) ; B ( − 1 ; 2) ; C ( − 2 ; 5 ; − 2 ; 5) 1 Place points A , B , and C . The graph will be completed as the questions in the exercise are answered. 2 a Determine the lengths AC and BC . b We assume that the segment [ AB ] has length 5 . Prove that triangle ABC is a right triangle at C . 3 Let K be the midpoint of segment [ AB ] . a Show that point K has coordinates : K ( − 2 ; 5 ; 0) . b Determine the length KC . c Draw the circle C with center K and passing through point A . https://chingmath.fr chapExoCorrec/6486 sacados/6486 -8-7-6-5-4-3-2-1234567I-4-3-2-12345JOA1B1A2B2A3B3A4B4 chapExoCorrec/6481 sacados/6481 IJO
ABu A(d123456789123456789(‹ E.9596 Definition: the length of each of its representatives is called the norm of a vector −→ u : Example: let’s consider the vector −→ u admitting as repre-sentative the vector −−→ AB as representative : The norm of the vector −→ u has the value : −→ u = AB Proposition: in the plane provided with an orthonormal reference frame, consider the vector −→ u ( x ; y ) . The norm of the vector −→ u has the value : −→ u = x 2 + y 2 Consider the plane provided with an orthonormal reference frame. 1 The vector −→ u has coordinates −→ u (5 ; 2) . Determine the norm of the vector −→ u . 2 Consider the two points A (4 ; 1) and B (0.5 ; 3) . Deter-mine the value of −−→ AB . E.7146 Characterizing properties of the parallelogram : Let ABCD be a quadrilateral. If the diagonals of ABCD intersect at their middles then ABCD is a parallelogram. If the opposite sides of ABCD are parallel two by two then ABCD is a parallelogram. If the opposite sides of ABCD are the same length then ABCD is a parallelogram. If two of the opposite sides are parallel and of the same length then ABCD is a parallelogram. Consider the following four points characterized by their co-ordinates in an ( O ; I ; J ) orthonormal coordinate system : A (2 ; 3) ; B ( − 2 ; 1) ; C ( − 4 ; − 3) ; D (0 ; − 1) Show that the quadrilateral ABCD is a parallelogram. E.6482 Property characterizing the rectangle : Let ABCD be a quadrilateral: If ABCD has three right angles then ABCD is a rect-angle. Let ABCD be a parallelogram: If ABCD has its diagonals of the same length then ABCD is a rectangle. If ABCD has a right angle then ABCD is a rectangle. Consider the following four points characterized by their co-ordinates in an ( O ; I ; J ) orthonormal coordinate system : A ( − 4 ; − 1) ; B ( − 3 ; − 4) ; C (3 ; − 2) ; D (2 ; 1) Show that the quadrilateral ABCD is a rectangle. E.10625 Definition: Let −→ u ( x ; y ) and −→ v ( x ; y ) be vectors. We call the de-terminant of vectors −→ u and −→ v , noted det( −→ u ; −→ v ) , defined by: det( −→ u ; −→ v ) = x × y − x × y two vectors −→ u and −→ v are said to be collinear if these two vectors have the same direction. Proposition: In a plane equipped with a coordinate sys-tem, consider the two vectors −→ u and −→ v . The two vectors −→ u and −→ v are collinear with each other if and only if their determinant is zero. In a plane equipped with an orthonormal coordinate system, consider the four points : A ( − 6 ; 3) ; B (2 ; − 1) ; C (4 ; − 1) ; D (10 ; − 4) 1 Determine the coordinates of vectors −−→ AB and −−→ CD . 2 Justify that vectors −−→ AB and −−→ CD are collinear. 2. Introduction E.9589 In the plane, consider the point A and the straight lines ( d ) , (Δ) , ( ‹ ) : 1 a Among the proposed points, note M the orthogonal project of the point A on the line ( d ) . b Among the proposed points, note N the orthogonal project of the point A on the line (Δ) . 2 Place the point P projected from the point A on the line ( ‹ ) . https://chingmath.fr chapExoCorrec/9596 sacados/9596 ABu chapExoCorrec/7146 sacados/7146 chapExoCorrec/6482 sacados/6482 sacados/10625 chapExoCorrec/9589 sacados/9589 A(d123456789123456789(‹
ABCDEF1u ¸ABCGH ABCHDEFH−−AB·−ACAB×AH−−DE·−−DF−DE×DH E.7432 In the plane, consider the six points and four vectors shown below : Note: the grid, points and unit have been chosen so that : AB = 8 u ; AC = 6 u ; DE = 6 u ; DF = 8 u Part A 1 a Represent the point M projected orthogonally from the point C onto the line ( AB ) . b Determine the value of the product : AB × AM 2 a Represent the point N projected orthogonally from the point B onto the line ( AC ) . b Determine the value of the product : AN × AC Definition: in the plane, consider three points A , B , C (we assume B distinct from A ) . Let H be the projected point C onto the line ( AB ) . We define the scalar product of the vectors −→ AB and −→ AC as the number defined by: AB × AH if the vecteurs −−→ AB and −−→ AH sont colinéaires and the same direction − AB × AH if the vecteurs −−→ AB and −−→ AH sont colinéaires and opposite directions. We note this number −−→ AB · −→ AC 3 What can we say about −−→ AB · −→ AC and −→ AC · −−→ AB ? Part B 4 Show that : −−→ DE · −−→ DF = − 24 5 Justify that : −−→ DE · −−→ DF = −−→ DF · −−→ DE E.8439 We consider three points A , B , C distinct two by two represented below : Let G (resp. H ) be the orthogonal project of the point C (resp. B ) onto the line ( AB ) (resp. ( AC ) ) : 1 a In the triangle AGC rectangular in G , give the ex-pression for cos ¸ . b In the triangle ABH rectangle in H , give the expres-sion for cos ¸ . 2 Deduce the equality: −−→ AB · −→ AC = −→ AC · −−→ AB 3. Scalar product and projection E.9588 Definition: In the plane, consider three points A , B , C (we assume B distinct from A ) . Let H be the projected point C onto the line ( AB ) . We define the scalar product of the vectors −→ AB and −→ AC as the number defined by: AB × AH if the vecteurs −−→ AB and −−→ AH sont colinéaires and same direction − AB × AH if the vecteurs −−→ AB and −−→ AH sont colinéaires and opposite directions. We note this number −−→ AB · −→ AC Illustration : https://chingmath.fr chapExoCorrec/7432 sacados/7432 ABCDEF1u chapExoCorrec/8439 sacados/8439 ¸ABCGH chapExoCorrec/9588 sacados/9588 ABCHDEFH−−AB·−ACAB×AH−−DE·−−DF−DE×DH
ABCDEF1u ABCI3cm DEFGH5cm2;5cm ABCDIJKLO6cm2;5cm ABCDEFGH ABCDO In a grid, consider the six points below : Determine the value of the scalar products : a −−→ AB · −→ AC b −−→ DE · −−→ DF E.8440 In the plane, consider the equilat-eral triangle ABC shown below and the point I midpoint of the seg-ment [ BC ] . Determine the following scalar products : a −−→ AB · −→ AC b −−→ BA · −→ BI E.9587 In the plane, consider the rect-angle DEFG where the point H is the middle of the diago-nal [ DF ] , determine the scalar products : a −−→ DF · −−→ DE b −−→ DG · −−→ DE c −−→ DF · −−→ HD E.9604 Consider the rectangle ABCD such that AB =6 cm and CB =2.5 cm : The points I , J , K , L are the respective middles of the sides [ AB ] , [ BC ] , [ CD ] , [ DA ] . The point O is the center of the rectangle ABCD . Determine the value of the following scalar products : a −→ AI · −→ AC b −→ IA · −→ IB c −→ IA · −→ BI d −→ DL · −−→ DO e −−→ KO · −−→ DB f −−→ AB · −−→ OC 4. Orthogonality and collinearity E.9590 In the plane, consider the three squares of side 2 shown below : Establish the following equalities: a −−→ AH · −−→ AB = 0 b −−→ BC · −−→ BC = 4 c −−→ FE · −−→ FH = − 8 E.9591 Proposition: let −→ u and −→ v be two vectors of the plane. If −→ u and −→ v are collinear: if −→ u and −→ v are the same way: −→ u · −→ v = −→ u × −→ v if −→ u and −→ v are in opposite directions : −→ u · −→ v = − −→ u × −→ v If −→ u and −→ v are orthogonal then : −→ u · −→ v = 0 Consider the square ABCD with side 1 and admitting point O as center represented below : Determine the scalar products : a −−→ BA · −−→ BC b −→ AO · −−→ OC c −−→ DO · −−→ CO d −−→ DC · −−→ BC 5. Scalar product and cosine https://chingmath.fr ABCDEF1u chapExoCorrec/8440 sacados/8440 ABCI3cm chapExoCorrec/9587 sacados/9587 DEFGH5cm2;5cm chapExoCorrec/9604 sacados/9604 ABCDIJKLO6cm2;5cm chapExoCorrec/9590 sacados/9590 ABCDEFGH chapExoCorrec/9591 sacados/9591 ABCDO
45o30oABCDE OIJ30o60o45oABCDEFGHK −F1¸−F2˛‚−F 1u−u−v¸−u−v¸−u−v¸abc E.2574 Proposition: For any triplet of points A , B , C that are distinct from each other, we have : −−→ AB · −→ AC = AB × AC × cos BAC Consider the figure below where : AE =4 cm and AC =2 cm and we equip the plane with an orthonormal coordinate system, oriented in the direct direction, whose unit measures 1 cm , and whose x-axis is the line ( AD ) . Determine the value of the scalar products below : a −−→ AB · −→ AE b −→ AC · −−→ AD c −−→ DA · −−→ DE Proposal: trigonometric table of notable angles : ¸ 0 π = 6 π = 4 π = 3 π = 2 cos ¸ 1 √ 3 = 2 √ 2 = 2 1 = 2 0 sin ¸ 0 1 = 2 √ 2 = 2 √ 3 = 2 1 tan ¸ 0 √ 3 = 3 1 √ 3 × E.2573 Consider the orthonormal reference frame ( O ; I ; J ) below : where the angles are given in radians and the two semicircles satisfy : OA =2 cm and OB =3 cm . 1 Justify that : −→ OI · −→ OJ = 0 2 Determine the exact values of the scalar products : a −→ OA · −−→ OC b −−→ OE · −−→ OB 3 Determine the scalar products, rounded to the nearest tenth : a −−→ OD · −−→ OE b −−→ OE · −−→ OH Proposition: Trigonometric table of notable angles : ¸ 0 π = 6 π = 4 π = 3 π = 2 cos ¸ 1 √ 3 = 2 √ 2 = 2 1 = 2 0 sin ¸ 0 1 = 2 √ 2 = 2 √ 3 = 2 1 tan ¸ 0 √ 3 = 3 1 √ 3 × E.3034 The diagram below shows a bal-anced pulley system. Each weight exerts a force on the knot proportional to its own weight. The following information is given : −→ F 1 = 8 N ; −→ F 2 = 6 N ; −→ F = 12 N We note R the resultant of all these forces : −→ R = −→ F + −→ F 1 + −→ F 2 1 Determine as a function of alpha , ˛ and ‚ the following three scalar products : −→ R · −→ F 1 ; −→ R · −→ F 2 ; −→ R · −→ F 2 We now assume that this system is in an equilibrium position, so we have −→ R = −→ 0 . a Show that the angle measures verify the following sys-tem : 4 · cos ¸ + 3 · cos ˛ + 6 = 0 6 · cos ¸ + 3 · cos ‚ + 4 = 0 6 · cos ˛ + 4 · cos ‚ + 3 = 0 b Deduce the values of ¸ , ˛ , ‚ for the equilibrium posi-tion. 6. Angle measurement and scalar product E.8441 Consider the three configurations, each with two vectors −→ u and −→ v : 1 For each question, determine the following values : https://chingmath.fr chapExoCorrec/2574 sacados/2574 45o30oABCDE chapExoCorrec/2573 sacados/2573 OIJ30o60o45oABCDEFGHK chapExoCorrec/3034 sacados/3034 −F1¸−F2˛‚−F chapExoCorrec/8441 sacados/8441 1u−u−v¸−u−v¸−u−v¸abc
A−u−v1u ABCDABFig.1Fig.2 ABCDABFig.1Fig.2 8cm10cm17cm6cm15cmABCD −→ u ; −→ v ; −→ u · −→ v 2 Determine the measure of the angle ¸ to the nearest tenth of a degree. 7. Discovering algebraic properties E.8442 Consider the two vectors −→ u and −→ v shown below : 1 a Place the points B and C such that : −→ u = −−→ AB ; −→ v = −→ AC b Determine the value of : −→ u · −→ v . 2 a Place the point D such that : 2 · −→ v = −−→ AD . b Determine the value of : −→ u · 2 · −→ v . 3 What relationship can be established? E.8445 In this exercise, we will check the validity of the identity below in special cases : −→ u · −→ v + −→ w To do this, consider 4 points A , B , C and D such that : −→ u = −−→ AB ; −→ v = −→ AC ; −→ w = −−→ AD To investigate the diversity of possible configurations, we should study 4 case disjunctions : only two are proposed here. Part A The projected values of the vectors −→ v and −→ w on the direction of the vector −→ u are in the same direction as the vector −→ u . 1 a Place the point H (resp. I ) orthogonal projected of the point C (resp. D ) on the line ( AB ) . b Determine the value of : −→ u · −→ v + −→ u · −→ w 2 a Place the point J verifying the relation: −→ AJ = −→ v + −→ w Place the point K orthogonally projected from the point J on the line ( AB ) . b Determine the value of : −→ u · −→ v + −→ w Part B The projected vector −→ v (resp. −→ w ) on the direction of the vec-tor −→ u is in the same direction (resp. in the opposite direction) than the vector −→ u . 3 a Place the point H (resp. I ) orthogonal projected of the point C (resp. D ) on the line ( AB ) . b Determine the value of : −→ u · −→ v + −→ u · −→ w 4 a Place the point J verifying the relation: −→ AJ = −→ v + −→ w Place the point K orthogonally projected from the point J on the straight line ( AB ) . b Determine the value of : −→ u · −→ v + −→ w Part C 5 What conjecture can be made about the two numbers : −→ u · −→ u + −→ v ; −→ u · −→ v + −→ u · −→ w 8. Use of algebraic properties, orthogonality and collinearity E.10629 Properties: let −→ u and −→ v be two vectors : −→ u · −→ v = −→ v · −→ u − −→ u · −→ v = − −→ v · −→ u −→ u · −→ v + −→ w = −→ u · −→ v + −→ u · −→ w Consider the two triangles ABC and ABD right-angled in B shown below with their measurements : 1 Verify equality: −−→ AD ·−−→ DB = − 225 https://chingmath.fr chapExoCorrec/8442 sacados/8442 A−u−v1u chapExoCorrec/8445 sacados/8445 ABCDABFig.1Fig.2 ABCDABFig.1Fig.2 sacados/10629 8cm10cm17cm6cm15cmABCD
12cm15cm37cm9cm35cmABCD −u−vABCDE ABCD8m9m15m 6;5cm6cm2;5cm5;6cm3;3cmABCD 2 Determine scalar products : a −−→ BA · −−→ DA b −−→ BC · −→ CA c −→ CA · −−→ DB E.10640 Consider the two triangles ABC and ABD right-angled in B shown below with their measure-ments : 1 Determine the scalar products : a −→ AC · −−→ AB b −→ AC · −−→ BD 2 Determine the value of the scalar product −→ AC · −−→ AD . E.10630 Proposition: let −→ u and −→ v be two vectors of the plane and – ∈ R . On a: – ×−→ u · −→ v = – · −→ u · −→ v Consider the plane provided with a paving, shown below, formed by equilateral triangles of side 3 and five points A , B , C , D , E : We note : −→ u = −−→ AB ; −→ v = −→ AC . 1 Determine the scalar product of the vectors −→ u and −→ v . 2 Determine the scalar products : a −−→ AD · −→ AC b −→ AE · −−→ AD E.9598 Proposition: let −→ u and −→ v be two vectors of the plane and for – ∈ R . −→ u · −→ v = −→ v · −→ u – ×−→ u · −→ v = – × −→ u · −→ v If −→ u and −→ v are orthogonal : −→ u · −→ v = 0 If −→ u and −→ v are collinear in the same direction : −→ u · −→ v = −→ u × −→ v If −→ u and −→ v are collinear in opposite directions : −→ u · −→ v = − −→ u × −→ v Consider the two triangles ABC and ABD rectangle in A and D respectively, shown below : 1 Establish that : BC =17 m ; BD =12 m 2 Determine the values of the following scalar products : a −−→ BC · −−→ BA b −−→ AB · −−→ BD c −−→ AD · −−→ DB d −−→ DB · −−→ AB e −−→ DA · −−→ AB f −−→ BC · −→ CA E.10641 Consider the two triangles ABC and ACD , which are right-angled at C and D respectively: 1 Determine the values of the scalar products : −−→ AD · −−→ AB ; −−→ AB · −→ AC 2 Deduce the value of the scalar product : −−→ AB ·−−→ CD 9. Decomposition and double-distribution E.9597 Reminders: −→ u · −→ v + −→ w = −→ u · −→ v + −→ u · −→ w −→ u + −→ v · −→ w + −→ t = −→ u · −→ w + −→ u · −→ t + −→ v · −→ w + −→ v · −→ t https://chingmath.fr sacados/10640 12cm15cm37cm9cm35cmABCD sacados/10630 −u−vABCDE chapExoCorrec/9598 sacados/9598 ABCD8m9m15m chapExoCorrec/10641 sacados/10641 6;5cm6cm2;5cm5;6cm3;3cmABCD chapExoCorrec/9597 sacados/9597
ABCD8cm4cmOIJ 45o30oABCDE ABCD3cm2cmx ABCDI ABCDIM¸5cm ABCDMIJ The rectangle ABCD is such that AB =8 cm and AD =4 cm . O is the center of the rectangle. Points I and J are the mid-points of sides [ AD ] and [ BC ] , respectively. 1 Establish that : −−→ OC · −−→ OD = − 12 2 Using the Pythagorean theorem, show that : OC = 2 5 3 Determine the oriented angle COD rounded to the near-est degree. E.8443 Consider the figure below où : AE =4 cm and AC =2 cm and we provide the plane with the orthonormal frame of ref-erence, oriented in the direct direction, whose unit measures 1 cm , and whose abscissa axis is the straight line ( AD ) . 1 Determine the exact values of the side lengths of the tri-angles ABC and ADE . 1 Establish equality: −−→ AD + −−→ DE · −−→ AB + −−→ BC = AD × AB − DE × BC 2 Determine the value of the scalar product : −→ AE · −→ AC E.8214 Consider the trapezoid ABCD shown below : where : AC =2 cm ; CD =3 cm Determine the length x of the segment [ AB ] so that the di-agonals, [ AD ] and [ BC ] , of the trapezoid ABCD are perpen-dicular. E.2665 Let a be a positive real number. Consider the rectangle ABCD such that : AB = a ; AD = 2 2 · a We note I the middle of [ CD ] Using algebraic properties alone, demonstrate that the straight lines ( AC ) and ( BI ) are perpendicular. E.9796 Establish the following property: ˇ In any parallelogram, the sum of the squares of the lengths of the sides is equal to the sum of the squares of the lengths of the diagonals. ı E.3014 In the plane, consider the rectangle ABCD such that : AB = 5 cm ; BC = 2 3 · AB I is the middle of the segment [ AB ] ; the straight lines ( AC ) and ( ID ) intercept at the point M . 1 Expressing the vectors using −−→ AD and −−→ AB , determine the value of the scalar product −→ ID · −→ AC 2 a Determine the lengths of segments [ DI ] and [ AC ] . b Deduce the measure of the angle IMC to the nearest tenth of a degree. E.2673 Consider the square ABCD below. M is a point belonging to the diagonal [ BD ] . Let I be the orthogonal project of M onto ( DC ) and J the orthogonal project of M onto [ BC ] . 1 Establish the following relationship : −→ DI · −−→ DC = −−→ BC · −→ JC 2 Deduce that the straight lines ( AI ) and ( DJ ) are per-pendicular. https://chingmath.fr ABCD8cm4cmOIJ chapExoCorrec/8443 sacados/8443 45o30oABCDE chapExoCorrec/8214 sacados/8214 ABCD3cm2cmx chapExoCorrec/2665 sacados/2665 ABCDI chapExoCorrec/9796 sacados/9796 chapExoCorrec/3014 sacados/3014 ABCDIM¸5cm chapExoCorrec/2673 sacados/2673 ABCDMIJ
ABCD4cm1;5cm6cm ABCH15cm12cm9cm13cm ABCD12cm9cm16cm ABCDKM¸ M6cm5cmxABCD E.10645 Consider the trapezoid ABCD shown below : where : AC =1.5 cm ; CD =4 cm ; AB =6 cm Determine the value of the scalar product of the diagonals of this trapezoid. E.9619 Consider the triangle ABC and H the foot of the height coming from the vertex B and whose measurements are shown below : 1 Establish that : −−→ BA · −−→ BC = 99 2 Deduce the measure of the angle ∠ ABC . E.10671 Consider the triangle ACD and note B the foot of the height from A . We have the measurements : AB = 12 cm ; BC = 9 cm ; BD = 16 cm Establish that the triangle ACD is a right-angled triangle in A . 10. Double distribution and condition on an angle E.9620 Consider a rectangle ABCD such that : AB = 5 cm ; AD = 3 cm Consider the point K belonging to the segment [ AB ] and such that AK = 2 cm . Let M be a point on segment [ DC ] , let x be the length MC . 1 Establish that : −−→ KM · −−→ KB = 9 − 3 x 2 Determine length KM as a function of x . 3 We wish to determine the position(s) of point M so that ∠ BKM =60 o . a Knowing that cos 60= 1 2 , show that the length x is so- lution of the equation : x 2 − 6 x + 6 = 0 b Deduce the position(s) of the point M satisfying ∠ BKM = 60 o . E.9618 In the plane, consider the rectan-gle ABCD and a point E on the segment [ DC ] such that : BC =5 cm ; AB =6 cm Let M be a point on the segment [ AD ] and let x be the length of the segment [ AM ] . 1 Determine an expression for the scalar product −−→ MB · −−→ MC in terms of x . 2 Determine the value(s) of x so that the angle BMC mea-sures : EMB =45 o Hint: use the factorization : x 4 − 10 x 3 +107 x 2 − 30 x +756 = x 2 − 5 x +6 x 2 − 5 · x +66 11. Orthogonality, collinearity, angle calculation E.8444 Consider the rectangle ABCD shown below where I is the point of intersection of its di-agonals and where the following dimensions are given : AB = 6 cm ; BC = 2 cm https://chingmath.fr chapExoCorrec/10645 sacados/10645 ABCD4cm1;5cm6cm chapExoCorrec/9619 sacados/9619 ABCH15cm12cm9cm13cm chapExoCorrec/10671 sacados/10671 ABCD12cm9cm16cm chapExoCorrec/9620 sacados/9620 ABCDKM¸ chapExoCorrec/9618 sacados/9618 M6cm5cmxABCD chapExoCorrec/8444 sacados/8444
ABCDIH ABC6cm2cm7cm ABCD5cm4cm3cm 1 Establish the following equality: −→ ID · −→ IC = 1 4 · AD 2 − 1 4 · AB 2 = − 8 2 a Determine the length of segment [ IC ] . b Deduce the measure of the angle ∠ DIC . 12. Scalar product and parallelogram E.5059 Proposition - Definition: Let −→ u and −→ v be two vec-tors in the plane equipped with an orthonormal coordinate system. We have the following identities: −→ u · −→ v = 1 2 −→ u 2 + −→ v 2 − −→ u − −→ v 2 −→ u · −→ v = 1 2 −→ u + −→ v 2 − −→ u 2 − −→ v 2 −→ u + −→ v 2 + −→ u − −→ v 2 = 2 · −→ u 2 + 2 · −→ v 2 These identities are called the identities of the parallel-ogram . Consider the triangle ABC such that : AB = 2 cm ; AC = 6 cm ; BC = 7 cm 1 Using the formula : −→ u − −→ v 2 = −→ u 2 + −→ v 2 − 2 ×−→ u · −→ v Determine the value of the scalar product : −−→ AB · −→ AC 2 a Place the point D such that the quadrilateral ABDC is a parallelogram. b Using the formula : −→ u + −→ v 2 = u 2 + v 2 + 2 ×−→ u · −→ v Determine the length of the diagonal [ AD ] rounded to the nearest millimeter. E.3011 1 For any vector −→ u and −→ v , establish the following equal-ity: −→ u + −→ v 2 − −→ u − −→ v 2 = 4 ×−→ u · −→ v 2 Consider the parallelogram ABCD in the plane. Note: −→ u = −−→ AB ; −→ v = −−→ BC a What do the vectors −→ u + −→ v and −→ u − −→ v represent for the parallelogram ABCD ? b Using the previous questions, establish the following proposition : ˇIn a parallelogram, diagonals are of equal length if, and only if, adjacent sides are perpendicular.ı E.3015 In the plane, consider the parallelo-gram ABCD having the following measures : AB = 5 cm ; AC = 4 cm ; AD = 3 cm 1 Recall the parallelogram formula : a Expand expression : −→ u + −→ v 2 . b Deduce the value of −−→ AB · −−→ AD as a function of vector norms. 2 a Expand the expression : −−→ AB − −−→ AD 2 . b Deduce the measure of the diagonal [ BD ] . 13. Coordinates and scalar product E.7781 Proposition: In the plane equipped with an orthonor-mal coordinate system O ; −→ i ; −→ j , consider the two vectors −→ u ( x ; y ) and −→ v ( x ; y ) . The scalar product of vectors −→ u and −→ v is a number denoted by −→ u · −→ v defined by: −→ u · −→ v = x · x + y · y Vectors −→ u and −→ v are orthogonal if and only if their scalar product is zero. Consider the plane equipped with a coordinate system O ; I ; J and the three points : A ( − 5 ; 1) ; B ( − 3 ; − 5) ; C ( − 2 ; 2) . Show that triangle ABC is a right triangle at A . E.9593 Consider the plane provided with a reference frame O ; I ; J orthonormal and the three points A (2 ; 1) , B (1 ; − 2) and C ( − 1 ; 2) . Justify that the triangle ABC is right-angled at A . https://chingmath.fr ABCDIH chapExoCorrec/5059 sacados/5059 ABC6cm2cm7cm chapExoCorrec/3011 sacados/3011 chapExoCorrec/3015 sacados/3015 ABCD5cm4cm3cm chapExoCorrec/7781 sacados/7781 chapExoCorrec/9593 sacados/9593
−i−jABCDI E.9615 In the plane with an orthonor-mal reference frame, consider the three points : A (2 ; 3) ; B (4 ; 7) ; C (7.6 ; 0.2) Justify that the triangle ABC is rectangular in B E.3018 In the plane provided with a O ; I ; J orthonormal coordinate system, consider the fol-lowing four points : A ( − 3 ; 2) ; B ( − 2 ; − 2) ; C (2 ; − 1) ; D (1 ; 3) 1 Determine the value of −−→ AB · −−→ AD 2 Demonstrate that the quadrilateral ABCD is a rectan-gle. E.9592 Consider the plane with a refer-ence frame O ; I ; J and the three points : D ( − 3 ; − 2) ; E (1 ; 1) ; F 2 ; − 26 3 . Show that the triangle DEF is right-angled. Specify the ver-tex of the right angle. E.8212 Consider the plane provided with a reference frame O ; I ; J orthonormal and the three points D ( − 1 ; 3) , E 3 ; 14 3 and F − 1 6 ; 1 . Justify that the triangle DEF is right-angled. E.7787 In the plane provided with a O ; I ; J orthonormal coordinate system, consider the three points : A ( − 2 ; 1) ; B ( − 8 ; − 3) ; D − 3 ; 5 2 1 Determine the coordinates of the point C such that the quadrilateral ABCD is a parallelogram. 2 Show that the quadrilateral ABCD is a rectangle. E.3013 Let a be a positive real num-ber. Consider the rectangle ABCD such that : AB = a ; AD = 2 2 a Let I be the midpoint of [ CD ] . A representation is given below : Consider the plane with an orthonormal coordinate system D ; −→ i ; −→ j ) in the direct direction where −→ i = −−→ DC : 1 Determine the coordinates of the various points in this figure. 2 Deduce that the straight lines ( AC ) and ( IB ) are per-pendicular. Subsidiary question : repeat question 2 without using the coordinates of the points. 14. Coordinates and finding the coordinates of a point E.8432 In a reference frame O ; I ; J , con-sider the two points A ( − 2 ; 3) and B (4 ; − 1) and a point C such that : the point C has abscissa 3 . triangle ABC is right-angled at B . Determine the coordinates of point C . E.9654 In a reference frame O ; I ; J , con-sider the two points B (2.2 ; − 0.5) and C (3.7 ; − 0.9) . The point A is such that : triangle ABC is right-angled at B ; the ordinate of point A has value 1 . Determine the coordinates of point A . E.9617 In a reference frame O ; I ; J , con-sider the two points A ( − 2 ; 3) and B (4 ; − 1) and a point C such that : the point C has abscissa 3 . triangle ABC is right-angled at B . Determine the coordinates of point C . E.5154 In a reference frame O ; I ; J , con-sider the two points A ( − 2 ; 3) and B (4 ; − 1) and a point C such that : the point C has abscissa 3 . triangle ABC is right-angled at C . Determine the coordinates of point C . E.5155 In a reference frame O ; I ; J , con-sider the two points A (2 ; 2) and B (4 ; − 4) and a point C such that : the point C has abscissa 3 . triangle ABC is right-angled at C . Determine the coordinates of point C . https://chingmath.fr chapExoCorrec/9615 sacados/9615 A reouvrir apres interro chapExoCorrec/3018 sacados/3018 chapExoCorrec/9592 sacados/9592 chapExoCorrec/8212 sacados/8212 chapExoCorrec/7787 sacados/7787 chapExoCorrec/3013 sacados/3013 −i−jABCDI chapExoCorrec/8432 sacados/8432 chapExoCorrec/9654 sacados/9654 chapExoCorrec/9617 sacados/9617 chapExoCorrec/5154 sacados/5154 chapExoCorrec/5155 sacados/5155
IJOCABMN IJOABCH E.9595 In the plane provided with an or-thonormal reference frame, consider the circle C of diam-eter [ AB ] and whose coordinates are known : A ( − 3 ; 0) ; B (2 ; 2) Determine the coordinates of the two points M and N inter-section of the circle C with the y-axis. Hint: we will use the following accepted proposition : Proposition: let C be a circle of diameter [ AB ] . for any point M of C distinct from A and B , the triangle ABM is a right-angled triangle at M . E.9599 In the plane provided with an or-thonormal reference frame, consider the points : A ( − 1 ; 1) ; B (3 ; 0) ; C (2 ; 2) Determine the coordinates of the projected point C on the line ( AB ) . 15. Norm of a vector E.8433 Definition: be −→ u a vector, we call norm of the vector −→ u its length and we note it −→ u . Proposition: in the plane provided with an orthonormal reference frame, the vector −→ u ( x ; y ) has norm : −→ u = x 2 + y 2 Consider the plane provided with a reference frame O ; I ; J orthonormal, the vector −→ u (3 ; 2) and the two points A (2 ; − 1) and B (4 ; 2) 1 Determine the norm of the vector −→ u . 2 Determine the norm of the vector −−→ AB . E.8434 1 Consider the vector −→ u − 5 2 ; 4 3 . Show that : −→ u = 17 6 2 Consider the two points A − 2 3 ; 1 2 and B 8 3 ; − 2 . Show that : −−→ AB = 25 6 E.8435 In the plane provided with a ref-erence frame O ; I ; J , consider the two points A , ordinate 2 5 and B 5 20 ; − 4 5 such that : −−→ AB = 5 4 . Determine the abscissa of point A . 16. Calculating angles in a reference frame E.8527 Proposition: In the plane provided with a reference frame O ; I ; J , consider three points ABC . We have the equal-ity: −−→ AB · −→ AC = AB × AC × cos ∠ BAC Consider the plane provided with an orthonormal reference frame ( O ; I ; J ) : Let A , B , C be three points in the plane with respective co-ordinates ( − 2 ; 3) , (1 ; − 4) and (0 ; − 2) 1 Determine the values of −−→ BA · −−→ BC , −−→ BA and −−→ BC . 2 Deduce the measure of the geometric angle ∠ ABC to the nearest hundredth of a degree. E.2596 Consider the plane provided with an orthonormal reference frame ( O ; I ; J ) and the following three points and their coordinates in this reference frame : A (3 ; 2) ; B (5 ; − 1) ; C ( − 2 ; 3) 1 Give the coordinates of the vectors −−→ AB , −→ AC and −−→ BC . 2 Give the values of the following scalar products : −−→ AB · −→ AC ; −−→ BA · −−→ BC ; −−→ CB · −→ CA 3 Determine distances AB , AC and BC . 4 Determine the measure of the 3 angles of the triangle ABC rounded to the nearest degree. https://chingmath.fr chapExoCorrec/9595 sacados/9595 IJOCABMN chapExoCorrec/9599 sacados/9599 IJOABCH chapExoCorrec/8433 sacados/8433 chapExoCorrec/8434 sacados/8434 chapExoCorrec/8435 sacados/8435 chapExoCorrec/8527 sacados/8527 chapExoCorrec/2596 sacados/2596
abABCDIJ AMPR E.2593 Consider the plane provided with an orthonormal reference frame ( O ; I ; J ) . Determine a measure of the angle oriented EDF where D (3 ; 5) , E ( − 1 ; 0) , F (2 ; 4) to the nearest hundredth of a de-gree. E.7849 Consider the plane provided with the reference frame O ; −→ u ; −→ v orthonormal and the points A , B , C with coordinates : A (1 ; 1) ; B (4 ; 2) ; C (3 ; − 1) Determine the measure of the angle ∠ ABC to the nearest tenth of a degree. E.9616 In the plane provided with an orthonormal reference frame, consider the three points : A (6 ; 3) ; B (1 ; 1) ; C (3 ; − 1) Determine, to the nearest tenth of a degree, the measure of the angle ∠ ACB . 17. Scalar product and algebraic manipulations E.3016 Consider the plane provided with an O ; I ; J orthonormal coordinate system and the follow-ing three points : A (2 ; 3) ; B (6 ; 5) ; C (0 ; 6) We note : −→ u = −−→ AB ; −→ v = −→ AC 1 a Determine −→ u and −→ v standards. b Determine the value of : −→ u · −→ v 2 a Expand expression : 3 ×−→ u − 2 ×−→ v 2 . b Deduce the norm : 3 ×−→ u − 2 ×−→ v . E.3081 Consider, in the plane, the rectangle ABCD of length a and width b ; let J and I be the orthogonal projects on the line ( AC ) of the points D and B respectively: 1 a Justify the following equality: −→ AC · −−→ BD = − AC × IJ b Justify the following equality: −→ AC · −−→ BD = b 2 − a 2 2 Deduce the expression for length IJ as a function of a and b . 18. Al-Kashi formula: determining a length E.8528 Consider the triangle ABC whose measures are: AC = 3.7 cm ; BC = 7 cm ; ∠ ACB = 48 o Determine the measurement, to the nearest millimetre, of seg-ment [ AB ] . E.6687 Proposal: Al-Kashi’s formulas applied to triangle ABC : AB 2 = AC 2 + BC 2 − 2 × AC × BC × cos ACB AC 2 = AB 2 + BC 2 − 2 × AB × BC × cos ABC BC 2 = AB 2 + AC 2 − 2 × AB × AC × cos BAC Consider the configuration below : 1 Write Al-Kashi’s three formulas in triangle ARP . 2 Copy and complete the dotted lines below : RP 2 = : : : 2 + : : : 2 − 2 × : : : × : : : × cos RMP E.8529 Consider a triangle ABC verifying the measures : AB = 5 cm ; AC = 3 cm ; ∠ ABC = 30 o Determine the possible measurements of segment [ BC ] achiev-ing these conditions. (these measurements will be given to the nearest millimetre) E.9655 Consider a triangle ABC ver-ifying the measures : BC = 17 cm ; AC = 23 cm ; ∠ BAC = 45 o Determine the possible measurements of segment [ AB ] achiev-ing these conditions. (these measurements will be given to the nearest millimetre) 19. Al-Kashi formula: determining an angle https://chingmath.fr chapExoCorrec/2593 sacados/2593 chapExoCorrec/7849 sacados/7849 chapExoCorrec/9616 sacados/9616 chapExoCorrec/3016 sacados/3016 chapExoCorrec/3081 sacados/3081 abABCDIJ chapExoCorrec/8528 sacados/8528 chapExoCorrec/6687 sacados/6687 AMPR chapExoCorrec/8529 sacados/8529 chapExoCorrec/9655 sacados/9655
ABCDO35o70o E.2590 Proposal: Al-Kashi’s formulas applied to triangle ABC give : AB 2 = AC 2 + BC 2 − 2 × AC × BC × cos ACB AC 2 = AB 2 + BC 2 − 2 × AB × BC × cos ABC BC 2 = AB 2 + AC 2 − 2 × AB × AC × cos BAC Consider triangle ABC , whose measurements are: AB = 5 ; 3 cm ; AC = 3 ; 7 cm ; BC = 7 cm Determine the measurement, to the nearest tenth of a degree, of the angles of triangle ABC . E.6706 Consider triangle ABC with the fol-lowing measurements : AB = 6.4 cm ; AC = 4.8 cm ; BC = 8 cm Determine the measurements of the three angles of triangle ABC , rounded to the nearest tenth of a degree. E.7850 Consider the triangle ABC whose measures are: AB = 5.5 cm ; AC = 6.2 cm ; BC = 4.7 cm Determine the measure of the angle ∠ BAC to the nearest tenth of a degree. E.7916 A craftsman wants to indus-trialize his manufacture of earrings (see below) . To do this, we need to help the technician from the chosen company determine the three angles of the triangle used as the geometric shape of the jewelry, as the manufacture of the cutting tool requires it. Angle measurements will be given in degrees, to an accuracy of 10 − 1 . E.10168 Consider the figure below où ABCD is a parallelogram. Determine the measure of the angle ∠ OAB . TO FINISH 20. Characterization of circle points E.8437 Let A and B be two distinct points in the plane. Let I be the midpoint of segment [ AB ] . 1 Establish, for any point M of the plane, the relationship : −−→ MA · −−→ MB = −−→ MI 2 − −→ AI 2 2 Consider a point C such that the triangle ABC is rect-angular at C . a Establish that : IC = IB = IA b What can be said about the point C relative to the circle C of diameter [ AB ] ? 3 Reciprocally, what can be said of the triangle ABM if the point M belongs to the circle C of diameter [ AB ] ? Establish this property. E.8436 In a reference frame O ; I ; J or-thonormal, consider the points A and B with coordinates : A ( − 2 ; 3) ; B (3 ; 0) and the circle C of diameter AB . Determine the coordinates of the two points on the circle C with abscissa 1 . The following proposition may be used : Proposition: if a triangle is inscribed in a circle and one of its sides forms a diameter then this triangle is right-angled and this side is its hypothenuse. 21. Further study: Sine formula E.9594 Consider the configuration below : https://chingmath.fr chapExoCorrec/2590 sacados/2590 chapExoCorrec/6706 sacados/6706 chapExoCorrec/7850 sacados/7850 chapExoCorrec/7916 sacados/7916 From Gilles Lab sacados/10168 ABCDO35o70o chapExoCorrec/8437 sacados/8437 chapExoCorrec/8436 sacados/8436 chapExoCorrec/9594 sacados/9594
AMPR ABCD48o35o85o6cm 5km3;5kmPMCB25o45o 4km2;5kmPMCB30o50o 1 Write the formula for equality of sines in the triangle RPM 2 Complete the equality: sin ∠ ARP = : : : × sin ∠ RAP : : : E.2674 Consider the quadrilateral ABCD shown below : 1 AL-Kashi’s formulas give the formula : DC 2 = BD 2 + BC 2 − 2 × BD × BC × cos ∠ DBC Deduce the measure of length DC rounded to the nearest millimetre. 2 The formula for sines expressed in the triangle ABD is expressed as : sin ∠ DBA AD = sin ∠ ADB AB = sin ∠ DAB DB Deduce the measures of lengths AB and AD rounded to the nearest millimetre. E.2664 A boat B reaches the harbor P in a straight line; on the edge of the bank, Marc and Cléa watch the boat returned to port. 1 a Determine the measures of the angles of the triangle BCM . b The sine formula is expressed in the triangle MBC by: sin ∠ BCM BM = sin ∠ CMB CB = sin ∠ MBC MC Deduct the length BC rounded to the nearest hectome-ter. 2 In the triangle CBP , Al-Kashi’s formulas are expressed as : PC 2 = PB 2 + BC 2 − 2 × PB × BC × cos ∠ PBC PB 2 = PC 2 + BC 2 − 2 × PC × BC × cos ∠ PCB CB 2 = CP 2 + PB 2 − 2 × CP × PB × cos ∠ CPB Deduce the distance separating the boat from the port, rounded to the nearest hectometre. E.6710 A boat B reaches the harbor P in a straight line; on the edge of the bank, Marc and Cléa watch the boat returned to port. Lengths will be rounded to the nearest hundred meters. 1 In the triangle MCB , determine the length BC . 2 Deduce the distance separating the boat from the port. https://chingmath.fr AMPR chapExoCorrec/2674 sacados/2674 ABCD48o35o85o6cm chapExoCorrec/2664 sacados/2664 5km3;5kmPMCB25o45o chapExoCorrec/6710 sacados/6710 4km2;5kmPMCB30o50o
3kmABM30o70oN95o35o ABCD52o47o85o6cm 3cmABCI20o60o75oDJ40o85oK115o BAC E.3084 Two observers wish to measure the distance separating the two lighthouses present near their coast. To do this, they separate by 3 km and take the fol-lowing angle measurements : ∠ MAB = 30 o ; ∠ MBA = 70 o ; ∠ NAB = 95 o ; ∠ ABN = 35 o The diagram below represents this situation : 1 a Determine the length of segment [ AN ] (to the near-est metre) . b Determine the length of segment [ AM ] (to the nearest metre) . 2 Determine the length of segment [ MN ] (to the nearest hectometer) . E.6707 Determine the measures of the four sides of the ABCD quadrilateral to the nearest millimetre. E.3041 Consider the configuration below where the line ( AK ) intercepts the segments [ BC ] and [ DC ] at I and J respectively: Determine, to the nearest millimetre, the length AK . (inter-mediate results must have an accuracy of 10 − 3 cm ) . Note: this method known as ˇ by triangularization ı of dis-tances was very useful in the Middle Ages for determining land and sea distances. 22. Further study: Remarkable lines and concurrency E.2661 Let ABC be any triangle. 1 Show that for any point M of the plane, we have the relation: −−→ AM · −−→ BC + −−→ BM · −→ CA + −−→ CM · −−→ AB = 0 2 Deduce that the heights of the triangle ABC are concur-rent at a point H . E.8438 Consider the triangle ABC shown below : Let I be the midpoint of the segment [ AB ] and define the point G by the vector relation: −→ IG = 1 3 · −→ IC 1 a Place the point G in the figure above. b Justify that the point G belongs to the median of the triangle ABC originating from the vertex C . 2 a Establish that the point G verifies the vector rela-tion : −→ GA + −−→ GB + −−→ GC = −→ 0 b Reciprocally, show that the point G is the only point M of the plane verifying the vector relation: −−→ MA + −−→ MB + −−→ MC = −→ 0 3 We note J the middle of the segment [ AC ] and H the point of the plane defined by the relation: −−→ JH = 1 3 · −→ JB a Show that the point H verifies the vector relation: −−→ HA + −−→ HB + −−→ HC = −→ 0 b What can be said about the point H ? Justify your answers. 4 Similarly, show that the point G belongs to the median of the triangle ABC originating from the vertex A . https://chingmath.fr chapExoCorrec/3084 sacados/3084 3kmABM30o70oN95o35o chapExoCorrec/6707 sacados/6707 ABCD52o47o85o6cm chapExoCorrec/3041 sacados/3041 3cmABCI20o60o75oDJ40o85oK115o chapExoCorrec/2661 sacados/2661 Concourances des hauteurs chapExoCorrec/8438 sacados/8438 BAC
23. Scalar product and sequences E.11044 Consider the two sequences u n and v n defined on N by: the suite u n is arithmetic with first term − 6 and reason 3 . the sequence v n is geometric with first term 27 and reason 1 3 . In the orthogonal plane, consider the sequence of points M n defined on N by: M n ( u n ; v n ) Demonstrate that the vectors −−−→ OM 2 and −−−→ OM 4 are orthogonal. E.11045 L’exercice n’existe pas. 24. Scalar product and derivative number E.11046 Consider the two functions : f ( x ) = x 2 − 5 x + 2 ; g ( x ) = − x 2 + 5 x − 1 Note C f and C g respectively, the representative curves of the functions f and g in an orthonormal frame of reference. Establish that the respective tangents to the curves C f and C g at the point of abscissa 3 are orthogonal. 25. Scalar product and exponential function E.11047 Consider the two functions f and g defined on R by: f ( x ) = e 2 x +2 ; g ( x ) = − 4 · e − 2 x +7 Note C f and C g the respective representative curves of the functions f and g in an orthonormal frame. For x a real number, let M x and N x be the two points with abscissa x and belong respectively to the curves C f and C g . Determine the set of values of x such that the two vectors −−−→ OM x and −−→ ON x are orthogonal. E.11048 L’exercice n’existe pas. 26. Unclassified financial years E.2662 Let ABC be a right-angled triangle at A . Note H the foot of the height from A . I , J , K are the respective middles of the segments [ AB ] , [ AC ] , [ BC ] . 1 Establish the following relationship : HA 2 = HB × HC 2 a Establish the following vector relationship : −→ AI + −→ AJ = −−→ AK b Demonstrate that the straight lines ( HI ) and ( HJ ) are perpendicular. E.7786 In a plane with a reference frame, consider the two straight lines ( d ) and (Δ) with equation : ( d ) : y = 3 · x − 1 ; (Δ) : 2 · x + 6 · y + 4 = 0 1 Demonstrate that the straight lines ( d ) and (Δ) are per-pendicular. 2 Determine the coordinates of the point of intersection of the straight lines ( d ) and (Δ) . https://chingmath.fr chapExoCorrec/11044 sacados/11044 sacados/11045 chapExoCorrec/11046 sacados/11046 chapExoCorrec/11047 sacados/11047 sacados/11048 chapExoCorrec/2662 sacados/2662 chapExoCorrec/7786 sacados/7786