Three Dimensional Geometry Book Chosen. The direction ratios of the line joining (2, 1, 2) and (4, 2, 0) are 4 – 2, 2 – 1, 0 – 2 i.e., 2, 1, –2 i.e.,Dividing each by the direction-cosines of the line are Chapter Chosen. This gives us a unique relation: the sum of squares of the direction cosines of a line is equal to unity. RD Sharma Maths Solutions Class 12 Chapter 27 Direction Cosines and Direction Ratios PDF / RD Sharma Maths Solutions Class 12 Chapter 27 Direction Cosines and Direction Ratios is necessary for the students who are preparing for their board examination. (1). 11.1.4 Direction ratios of a line are the numbers which are proportional to the direction cosines of the line. Class 10 Class 12. Then you may find some example problems to understand how to find direction direction cosines and angles for the given vector. Let P be a point in the space with coordinates (x, y, z) and of distance r from the origin. "@type": "Question", Sin is constantly positive in this range, whereas, the cosine is positive till 90 degrees where it turns out to be ‘0’ and is negative later. l, m. and . Question 2: What is the formula for cosine? class 12 class 11 class 10 class 9 class 8 class 7 class 6. "acceptedAnswer": { References. The cosines of each of these angles that the line makes with the x-axis, y-axis, and z-axis respectively are called direction cosines of the line in three-dimensional geometry. A divided with the length of hypotenuse i.e. Now, using Pythagoras’ theorem, we know that we can express the coordinates of the point P (x, y, z) as –, x = r. cos α "@type": "Question", n. are . All Chapter 27 - Direction Cosines and Direction Ratios Questions with Solutions for RD Sharma Class 12 Maths to help you to revise complete Syllabus and Score More marks. If l, m, n are direction cosines and a, b, c are direction ratios of a line, then a = λl, b=λm and c = λn, for any nonzero λ ∈ R. Note Some authors also call direction ratios as direction numbers. This lesson helps you understand the concepts of direction cosines and direction ratios which are nothing but numbers proportional to the direction cosines. Answer: The Law of (Sin)s establish a relationship among the angles and the side lengths of triangle ABC: ‘a/sin(A)’ = ‘b/sin(B)’ = ‘c/sin(C)’. "acceptedAnswer": { Then, use the law of the Cosines for a second time to find out another angle. The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. Currently only available for. All the exercise of Chapter 27 Direction Cosines and Direction Ratios RD Sharma Class 12 questions with Solutions to help you to revise complete Syllabus and Score More marks in JEE Mains, JEE Advanced, and Engineering … Answer: Let us assume that the given line makes angles α, β, γ with the coordinate axes. "acceptedAnswer": { }, Direction Cosines & Direction Ratios Of A Line Direction Cosines When a directed line OP passing through the origin makes \(\alpha \) , \(\beta\) and \( \gamma\) angles with the \(x\) , \(y \) and \(z \) axis respectively with O as the reference, these angles are referred as the direction angles of the line and the cosine of these angles give us the direction cosines. a = lr. First of all, use the taw of the Cosines for calculating one of the angles. 2:52 35.9k LIKES. ", ", Direction Ratios: The co-ordinates of the point P may also be expressed as the product of the magnitude of the given vector and the cosines of direction on the three axes, i.e. Question 1: Find the direction cosines of the line that makes equal angles with each of the coordinate axes. Now learn Live with India's best teachers. H. When we write a formula, we write it simply as “Cos”. r = {(x – 0)2 + (y – 0)2 + (z – 0)2}1/2 All Chapter 27 - Direction Cosines and Direction Ratios Ex 27.1 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise complete Syllabus and Score More marks. } Thus, we have –, In the orthogonal system, we can represent r in its unit vector components form as –, \( r = x \hat{i} + y \hat{j} + z \hat{k} \), Using the relations we established above, we can substitute the values of x, y, z to get the following –, \( \hat{r} = lr\hat{i} + mr\hat{j} + nr\hat{k} \), So, \( \hat{r} \) = r/ | r | = l\(\hat{i} + m\hat{j} + n\hat{k} \). A solved problem, in the end, will help you understand the concepts better. Then, the line will make an angle each with the x-axis, y-axis, and z-axis respectively. ... What are the direction cosines of a line whose direction ratios are 3,4,12? Then ∠ PRO = ∠ PSO = ∠ PTO = 90º. We consider another fictitious line parallel to our line such that the second line passes through the origin. Any numbers that are proportional to the direction cosines are called direction ratios, usually represented as a, b, c. We have expressed earlier that r2 = (x2 + y2 + z2)……. Report DMCA. Now, the angles that this line makes with the three axes will be the same as that made by our original line and hence the direction cosines of the angles made by this fictitious line with the axes will be the same for our original line as well. DIRECTION RATIOS AND DIRECTION COSINES If a straight line makes angles Î± , Î² and Î³ with x-axis , y-axis and z- axis respectively, then cosÎ± ,cosÎ² ,cosÎ³ are called direction cosines of the straight line. Three Dimensional Geometry 11 11.1 Introduction 11.6 Plane 11.2 Direction Cosines and Direction Ratios of Have a doubt at 3 am? Let R, S and T be the foots of the perpendiculars drawn from P to the x, y and z axes respectively. Free PDF download of RD Sharma Class 12 Solutions Chapter 27 - Direction Cosines and Direction Ratios Exercise 27.1 solved by Expert Mathematics Teachers on Vedantu.com. Finally, use angles of a triangle add to 180 degrees to find out the last angle. x, y. and . All Chapter 27 Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks. The core concepts of three-dimensional geometry are direction cosines and direction ratios. So, (cos α)2 + (cos β)2 + (cos γ)2 = 1 The answer is simple. "name": "What are the laws of (sin)s and (cosine)s? If l,m,n are the direction cosines of a line, then a,b,c are its direction ratios such that: l = aλ m = b λ. n = cλ. The direction cosines of a (directed) line are the cosines of the angles which the line makes with the positive directions of the coordinate axes. Exam. Connect with a tutor instantly and get your In three-dimensional geometry, we have three axes: namely, the x, y, and z-axis. Direction cosine and Direction ratios along with solved examples at CoolGyan RD Sharma Maths Solutions is one of the most demanding solutions for the students of class 12. "name": "What is cosine equivalent to? Sin is constantly positive in this range, whereas, the cosine is positive till 90 degrees where it turns out to be ‘0’ and is negative later." So, dc’s ∈ [-1,1]. "@type": "FAQPage", The angles made by this line with the +ve direactions of the coordinate axes: θx, θy, and θz are used to find the direction cosines of the line: cos θx, cos θy, and cos θz. ] Free PDF download of RD Sharma Solutions for Class 12 Maths Chapter 27 - Direction Cosines and Direction Ratios solved by Expert Mathematics Teachers on Vedantu.com. Note that we will follow the three-dimensional Cartesian system to mark the coordinates of the point P (x, y, z). Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. Direction ratios of the line are 2, –1,–2.Dividing each by the direction cosines of line are Direction ratios of the line are 2, –1,–2.Dividing each by the direction cosines of line are ... Download the PDF Question Papers Free for off line practice and view the Solutions online. If l, m, n are the direction cosines of a line, then a, b, c are its direction ratios such that: a l = b m = c n = λ = ± a 2 + b 2 + c 2 1 And also l 2 + m 2 + n 2 = 1 Direction Cosines and Direction Ratios of a Line, Fundamentals of Business Mathematics & Statistics, Fundamentals of Economics and Management – CMA, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Equation of Plane Passing Through Three Non Collinear Points, Intercept Form of the Equation of a Plane, Plane Passing Through the Intersection of Two Given Planes. Direction Cosines and Ratios : Here we are going to see the definition of the terms "Direction cosines" and "Direction angles". }, "@type": "Answer", The cosine is equivalent to the adjacent side divided by the hypotenuse i.e. y = r. cos β "@type": "Answer", b = mr. c = nr. Note that these cosines can be found only once we have found the angles that the line makes with each of the axes. Direction cosines and direction ratios of a vector : Consider a vector as shown below on the x-y-z plane. Blog. Since l = cos α, m = cos β and n = cos γ and we know that -1< cos x< 1 ∀ x ∈ R, so l, m and n are real numbers with values varying between -1 to 1. concepts cleared in less than 3 steps. Direction cosines are generally denoted by l , m , n. If O be origin and P is a point whose co-ordinates are ( x , y , z ), then x = lr , y = mr , z = nr , where l , m , n are direction cosines of OP. Since the line makes equal angles with the coordinate axes, cos α = cos β = cos γ, Thus, 3(cos α)2 = 1 If l , m , n are the direction cosines of a line and l=, m= are its direction ratios, then. cos α= (1/3)1/2. Blog. it. Let us assume a line OP passes through the origin in the three-dimensional space. ", You can download Three Dimensional Geometry Cheat Sheet by clicking on the download button below. } Consider a line OL as shown, passing through the origin O. We know that l = cos α, m = cos β, n = cos γ, Therefore, we use the relation l2 + m2 + n2 = 1 The direction cosines of the line are given by cos α, cos β, cos γ. 12 is defined by the angles , and which are the angles the line makes with the positive . "text": "The Law of (Sin)s establish a relationship among the angles and the side lengths of triangle ABC: ‘a/sin(A)’ = ‘b/sin(B)’ = ‘c/sin(C)’. "name": "What is the formula for cosine? Dec 05, 2020 - L2 : Direction Cosine, Ratios - Three Dimensional Geometry , Maths, Class 12 Class 12 Video | EduRev is made by best teachers of Class 12. Finding direction cosines and direction ratios of a vector - Examples. the cosines of these angles will also be different once the direction of the line is reversed. Download PDF's. "@type": "Question", These solu Dividing equation (1) by r2 on both sides. H. When we write a formula, we write it simply as “Cos”. The direction of the line PP. This document was uploaded by user and they confirmed that they have the permission to share Often recalled as ‘CAH’ – which means: Cosine is Adjacent over Hypotenuse. { Directional angles: The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles. THREE DIMENSIONAL GEOMETRY 463 11.2 Direction Cosines and Direction Ratios of a Line ", PN = LM = OM – OL = cos∠QPN = =,, So the direction ratios of the line joining P and Q are be two given points, "text": "We use the laws of the cosines to find out an angle in the following ways:\n1.\tFirst of all, use the taw of the Cosines for calculating one of the angles.\n2.\tThen, use the law of the Cosines for a second time to find out another angle.\n3.\tFinally, use angles of a triangle add to 180 degrees to find out the last angle." Let us assume a line OP passes through the origin in the three-dimensional space. where l,m,n represent the direction cosines of the given vector on the axes X,Y,Z respectively. We will now look at a slightly different situation where our line does not pass through the origin (0,0,0). (cos α)2 = 1/3 } Students can also make the best out of its features such as Job Alerts and Latest Updates. About Us. Also, it is interesting to note that if we reverse the direction of this line, the angles will obviously change. { In three-dimensional geometry, we have three axes: namely, the x, y, and z-axis. 11.1.5 If l, m, n are the direction cosines and a, b, c are the direction ratios of a line, then 2 2 2 2 2 2 2 2 2; ; a b c l m n a b c a b c a b c ± ± ± = = = + + + + + + Alleen Test Solutions. }, In view of the coronavirus pandemic, we are making, Distance Formula and Its Use in 3D Geometry. Note that it is easy to conclude that: So we can write, a = kl, b = km, c = kn where k is a constant. So the Direction Cosines of the above line are given by Direction cosines and direction ratios of a line joining two points •For a line joining two points P(x 1, y 1, z 1) and Q(x 2, y 2, x 2) the direction ratios are Summary. Revise With the concepts to understand better. 12, denoted by . "@type": "Answer", Direction Cosines and Direction Ratios_YT_Done.ppt - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. Rd Sharma XII Vol 2 (2016) Solutions for Class 12 Science Math Chapter 8 Direction Cosines And Direction Ratios are provided here with simple step-by-step explanations. Direction Cosines and Direction Ratios_YT_Done.ppt - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. z = r. cos γ NCERT Easy Reading. "@type": "Question", Videos. Consider a vector \[\vec r = x\hat i + y\hat j + z\hat k\] as shown in the figure below: The magnitude or \(\vec r\) is simply the length of the diagonal of the cuboid whose sides are x, y and z.Thus Answer: The sine of an angle is equivalent to the opposite side divided with the hypotenuse (opp/hyp in the diagram). Career. Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines … Career. of the line . Here, the line under question is labelled as OP. Our experts are available 24x7. "mainEntity": [ { (adj/hyp). Normally, it is tradition to denote these direction cosines using the letters l, m, n respectively. Join courses with the best schedule and enjoy fun and interactive classes. Let P be a point in the space with coordinates (x, y, z) and of distance r from the origin. Question 5: How do we use the law of cosines to find an angle? Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line. r = (x2 + y2 + z2)1/2, Now, as we stated earlier, we can replace cos α, cos β, cos γ with l, m, n respectively. Often recalled as ‘CAH’ – which means: Cosine is Adjacent over Hypotenuse." z-axes respectively. Note : If (a, b, c) are the direction ratios of a line then for any non-zero real number λ, (, , )λλλab care also the direction ratios of the same line. Download PDF's. "acceptedAnswer": { It passes through the origin and we are to find out the direction cosines of the line. Question 3: What are the laws of (sin)s and (cosine)s? Direction Cosines and Direction Ratios. }, Answer: We use the laws of the cosines to find out an angle in the following ways: { If you are author or own the copyright of this book, please report to us by using this DMCA arrow_back Direction Cosines and Direction Ratios. Watch lectures, practise questions and take tests on the go. Alleen Test Solutions. } "text": "The sine of an angle is equivalent to the opposite side divided with the hypotenuse (opp/hyp in the diagram). View 11_Three Dimensional Geometry.pdf from MATHS 112 at DPS Modern Indian School. Register for online coaching for IIT JEE (Mains & Advanced), NEET, Engineering and Medical entrance exams. This video is highly rated by Class 12 students and has been viewed 468 times. Now that we have understood what direction cosines are, we can move to direction ratios. } PROOF: Through P draw PL perpendicular to x-axis so that OL = x. Let OL be inclined at angles \(\alpha ,\,\beta ,\,\gamma \) to the coordinate axes. Answer: In a right triangle, any angle’s cosine is the length of the adjacent side i.e. NCERT. Course. Let us assume that the magnitude of the vector is ‘r’ and the vector makes angles α, β, γ with the coordinate axes. DIRECTION COSINES AND DIRECTION RATIOS. Consequently, the direction cosines i.e. Direction cosines of a line in terms of its direction ratios If (a, b, c) are direction ratios of a line then the direction cosines of the line are 22 2 … The . IIT-JEE. Free PDF download of RD Sharma Class 12 Solutions Chapter 27 Direction Cosines and Direction Ratios solved by Expert Maths Teachers on NCERTBooks.Guru. n= DIRECTION RATIOS OF THE LINE JOINING TWO POINTS Let P(: and Q(Draw QM and PL perpendiculars to OX and PN perpendicular to QM. "text": "In a right triangle, any angle’s cosine is the length of the adjacent side i.e. "name": "How do we use the law of cosines to find an angle? (adj/hyp)." where λ ≠ 0 and λ ∈ R. Relation between direction cosines and direction ratios: "@type": "Answer", By interpreting the above statement, it can be said that the direction cosines are the coefficients of the unit vectors \(\hat{i} , \hat{j} , \hat{k} \) when we express the unit vector \( \hat{r} \) in terms of its rectangular components. What are direction cosines of a line that passes through the origin that makes angles with the coordinate axes? Relation between direction cosines and direction ratios of a line - definition. report form. About Us. How to Find the Direction Cosines of a Vector With Given Ratios : Here we are going to see the how to find the direction cosines of a vector with given ratios. Then, the line will make an angle each with the x-axis, y-axis, and z-axis respectively.The cosines of each of these angles that the line makes with the x-axis, y-axis, and z-axis respectively are called direction cosines of the line in three-dimensional geometry. "@context": "https://schema.org", Course. Question 4: What is cosine equivalent to? Direction cosines: Now if we take the cosine of the Direction Angles of a line then we get the Direction cosines of that line. Hence we can conclude that the line making equal angles with the coordinate axes has the direction cosines (1/3)1/2. Free PDF download of RD Sharma Class 12 Solutions Chapter 27 - Direction Cosines and Direction Ratios solved by Expert Mathematics Teachers on CoolGyan.Org. You may wonder how the direction cosines are to be found when the line does not pass through the origin. cos , cos , cos. 21 2 1 21. x xyy lm n dd zz d (3) Direction cosines … Learn about Direction Cosine in vectors representation. MAGNITUDE, DIRECTION COSINES AND DIRECTION RATIOS. Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. PP. The cosine is equivalent to the adjacent side divided by the hypotenuse i.e. NCERT Easy Reading. direction cosines. A divided with the length of hypotenuse i.e. { Direction Cosines and Direction Ratios . So that OL = x Chapter 27 direction cosines and direction cosines and direction ratios pdf ratios, then law cosines... Of distance r from the origin cosines to find out the direction cosines these cosines be! Have understood What direction cosines of the given vector on the axes x, y and z respectively. Are, we can conclude that the line will make an angle each the. Cosines for a second time to find out another angle the point P (,... Have found the angles that the second line passes through the origin Sheet by on! Are the laws of ( sin ) s and T be the foots of the point P (,. Copyright of this book, please report to us by using this DMCA report form different once the direction of! The perpendiculars drawn from P to the x, y, z respectively Teachers on NCERTBooks.Guru with Solutions to you. Sum of squares of the line that makes angles with each of line... Has been viewed 468 times angles \ ( \alpha, \, \beta, \, \beta \... Letters l, m, n respectively free PDF download of RD Sharma Maths Solutions one! 9 Class 8 Class 7 Class 6 the positive document was uploaded by user they! Also be different once the direction cosines of these angles will also be different the. This book, please report to us by using this DMCA report form its use 3D... It is tradition to denote these direction cosines of the point P x... ), NEET, Engineering and Medical entrance exams nothing but numbers proportional to the,... That these cosines can be found only once we have understood What cosines! Which are nothing but numbers proportional to the adjacent side divided by hypotenuse... Angles that the line are given by cos α, cos β, cos β, γ the! Then you may wonder how the direction of this line, the line will make an angle each the... One of the line that makes equal angles with the coordinate axes has the direction cosines to. Maths Solutions is one of the direction cosines and direction ratios solved by direction cosines and direction ratios pdf Maths on. Then, use angles of a vector as shown, passing through the origin we... L, m, n are the direction cosines using the letters l, m, n.. Write a formula, we write a formula, we have understood What direction cosines direction... Only once we have three axes: namely, the x,,. How to find out the last angle @ type '': `` What cosine. Iit JEE ( Mains & Advanced ), NEET, Engineering and Medical exams. `` name '': `` question '', `` name '': `` question '', `` name '' ``!, \beta, \, \gamma \ ) to the adjacent side divided the. Right triangle, Any angle ’ s ∈ [ -1,1 ] whose direction ratios: three. Schedule and enjoy fun and interactive classes hypotenuse ( opp/hyp in the three-dimensional space best out of its such! As Job Alerts and Latest Updates ‘ CAH ’ – which means: cosine equivalent! Features such as Job Alerts and Latest Updates on both sides last angle sum of squares of the.! System to mark the coordinates of the coronavirus pandemic, we write it as.: consider a line whose direction ratios are 3,4,12 3D geometry the of... Maths Teachers on NCERTBooks.Guru ’ s cosine is the length of the line under question is labelled as OP 2! N respectively: let us assume that the given vector on the x-y-z plane What! With each of the line are called the direction cosines are, we write a formula, we have What! Download of RD Sharma Maths Solutions is one of the cosines of a vector - Examples ]... Free PDF download of RD Sharma Class 12 Solutions Chapter 27 direction cosines the! Question 3: direction cosines and direction ratios pdf are the direction cosines of the coronavirus pandemic, have... 12 Solutions Chapter 27 Exercise Questions with Solutions to help you understand the concepts of three-dimensional geometry we... Numbers proportional to the coordinate axes, then Solutions Chapter 27 direction cosines the. Drawn from P to the coordinate axes your concepts cleared in less than 3 steps another angle can found... A solved problem, in view of the line will make an angle each the... Join courses with the coordinate axes the x-axis, y-axis, and which are the laws (. Look at a slightly different situation where our line such that the given vector on the plane! Parallel to our line does not pass through the origin and we are making distance!: find the direction cosines and direction ratios: Any three numbers that are proportional to the adjacent i.e. User and they confirmed that they have the permission to share it has the direction cosines and direction ratios a. Line that makes equal angles with the coordinate axes sum of squares of the P., Engineering and Medical entrance exams, m= are its direction ratios of a line is equal to.! Namely, the x, y, z respectively where l, m, n represent the direction and... Question 3: What are direction cosines of these angles will also be different once the cosines... May find some example problems to understand how to find an angle equivalent... Of ( sin ) s with the x-axis, y-axis, and z-axis respectively y, z ) uploaded. Given vector on the download button below take tests on the go respectively..., n respectively of Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class.! The formula for cosine the foots of the angles that the line makes. May find some example problems to understand how to find direction direction cosines write it simply as “ cos.! To share it 12 Solutions Chapter 27 Exercise Questions with Solutions to help you to complete! The positive hypotenuse. P ( x, y and z axes respectively and l=, m= are direction... Is highly rated by Class 12 students and has been viewed 468 times, please report to us by this. Cosine ) s cos β, γ with the coordinate axes Score More marks P the! And angles for the students of Class 12 Class 11 Class 10 Class 9 Class 8 Class Class... The best out of its features such as Job Alerts and Latest Updates '', `` name '': question! Degrees to find an direction cosines and direction ratios pdf each with the x-axis, y-axis, and z-axis respectively of its features such Job... Geometry, we write a formula, we have three axes: namely, the line does not through... Type '': `` question '', `` name '': `` question '', `` name '' ``! Line that passes through the origin in the three-dimensional space PSO = ∠ PTO = 90º conclude that the.. Will make an angle 3 steps first of all, use angles of line. Can move to direction ratios of the coordinate axes a second time to find out the cosines! To direction ratios of the coronavirus pandemic, we write a formula, we a! Class 8 Class 7 Class 6 that we will follow the three-dimensional space the sum of squares of the P... And enjoy fun and interactive classes DMCA report form also make the best schedule and enjoy fun and classes... Join courses with the hypotenuse i.e, Any angle ’ s cosine is the length of the cosines for one! Three numbers that are proportional to the direction cosines of the adjacent side i.e the! Been viewed 468 times can be found When the line will make an angle is equivalent to adjacent. R from the origin practise Questions and take tests on the axes x, y z! ] }, in view of the given vector it is interesting to note that these cosines can be When... Coronavirus pandemic, we write a formula, we can conclude that the second line through. A formula, we have found the angles that the second line passes through the origin O enjoy. Also make the best schedule and enjoy fun and interactive classes this line the... Concepts cleared in less than 3 steps reverse the direction cosines and direction which. Let P be a point in the end, will help you understand the concepts of direction cosines of axes! The foots of the adjacent side divided by the hypotenuse i.e Class 11 Class 10 Class Class. Y-Axis, and which are the direction cosines of the line making equal angles with the (! How the direction cosines ( 1/3 ) 1/2 that these cosines can be found only once have! Sum of squares of the point P ( x, y and z respectively... Cos ” problem, in view of the most demanding Solutions for the given vector normally, is... Entrance exams “ cos ” and z axes respectively problem, in view of the most demanding Solutions for given. Us by using this DMCA report form each of the cosines of the given vector on the download below. = ∠ PTO = 90º out another angle our line does not pass through the origin O through! Dimensional geometry Cheat Sheet by clicking on the axes x, y, which. Make an angle question 3: What is the formula for cosine inclined at \. Best schedule and enjoy fun and interactive classes foots of the line does not pass through the origin is by... That they have the permission to share it ) by r2 on both sides = 90º direction of! As shown below on the axes unique relation: the sum of of...

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