direction cosines and direction ratios pdf

Register for online coaching for IIT JEE (Mains & Advanced), NEET, Engineering and Medical entrance exams. PROOF: Through P draw PL perpendicular to x-axis so that OL = x. ] (1). } n. are . Join courses with the best schedule and enjoy fun and interactive classes. Question 4: What is cosine equivalent to? Then ∠ PRO = ∠ PSO = ∠ PTO = 90º. Also, it is interesting to note that if we reverse the direction of this line, the angles will obviously change. Relation between direction cosines and direction ratios of a line - definition. "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)’. Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. Normally, it is tradition to denote these direction cosines using the letters l, m, n respectively. H. When we write a formula, we write it simply as “Cos”. 12, denoted by . 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)……. Hence we can conclude that the line making equal angles with the coordinate axes has the direction cosines (1/3)1/2. The cosine is equivalent to the adjacent side divided by the hypotenuse i.e. Connect with a tutor instantly and get your "acceptedAnswer": { We will now look at a slightly different situation where our line does not pass through the origin (0,0,0). Direction Cosines and Direction Ratios . 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. "@type": "Question", Note that these cosines can be found only once we have found the angles that the line makes with each of the axes. Now learn Live with India's best teachers. "text": "In a right triangle, any angle’s cosine is the length of the adjacent side i.e. Free PDF download of RD Sharma Class 12 Solutions Chapter 27 Direction Cosines and Direction Ratios solved by Expert Maths Teachers on NCERTBooks.Guru. "@type": "Answer", a = lr. Let P be a point in the space with coordinates (x, y, z) and of distance r from the origin. About Us. 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 … A divided with the length of hypotenuse i.e. 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. }, 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 Career. Report DMCA. Download PDF's. Often recalled as ‘CAH’ – which means: Cosine is Adjacent over Hypotenuse." This video is highly rated by Class 12 students and has been viewed 468 times. "@type": "Answer", Class 10 Class 12. "@type": "FAQPage", 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. cos , cos , cos. 21 2 1 21. x xyy lm n dd zz d (3) Direction cosines … The core concepts of three-dimensional geometry are direction cosines and direction ratios. "name": "How do we use the law of cosines to find an angle? RD Sharma Maths Solutions is one of the most demanding solutions for the students of class 12. In three-dimensional geometry, we have three axes: namely, the x, y, and z-axis. 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 The cosine is equivalent to the adjacent side divided by the hypotenuse i.e. ... What are the direction cosines of a line whose direction ratios are 3,4,12? First of all, use the taw of the Cosines for calculating one of the angles. 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. "acceptedAnswer": { arrow_back Direction Cosines and Direction Ratios. 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)’. }, Our experts are available 24x7. What are direction cosines of a line that passes through the origin that makes angles with the coordinate axes? report form. NCERT. 12 is defined by the angles , and which are the angles the line makes with the positive . Then, the line will make an angle each with the x-axis, y-axis, and z-axis respectively. Revise With the concepts to understand better. NCERT Easy Reading. ", Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. Consequently, the direction cosines i.e. Blog. 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\(\hat{i} + m\hat{j} + n\hat{k} \). H. When we write a formula, we write it simply as “Cos”. Let P be a point in the space with coordinates (x, y, z) and of distance r from the origin. "@type": "Question", Let us assume that the magnitude of the vector is ‘r’ and the vector makes angles α, β, γ with the coordinate axes. "acceptedAnswer": { 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. 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. You may wonder how the direction cosines are to be found when the line does not pass through the origin. 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 ± ± ± = = = + + + + + + Have a doubt at 3 am? In three-dimensional geometry, we have three axes: namely, the x, y, and z-axis. Answer: The sine of an angle is equivalent to the opposite side divided with the hypotenuse (opp/hyp in the diagram). Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines … Then, use the law of the Cosines for a second time to find out another angle. 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. The direction cosines of the line are given by cos α, cos β, cos γ. 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. { 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." Note that we will follow the three-dimensional Cartesian system to mark the coordinates of the point P (x, y, z). Direction cosine and Direction ratios along with solved examples at CoolGyan the cosines of these angles will also be different once the direction of the line is reversed. 2:52 35.9k LIKES. "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. View 11_Three Dimensional Geometry.pdf from MATHS 112 at DPS Modern Indian School. Course. Since the line makes equal angles with the coordinate axes, cos α = cos β = cos γ, Thus, 3(cos α)2 = 1 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. "@type": "Answer", It passes through the origin and we are to find out the direction cosines of the line. The answer is simple. Videos. Exam. This gives us a unique relation: the sum of squares of the direction cosines of a line is equal to unity. Let us assume a line OP passes through the origin in the three-dimensional space. Question 1: Find the direction cosines of the line that makes equal angles with each of the coordinate axes. Three Dimensional Geometry Book Chosen. Direction cosines and direction ratios of a vector : Consider a vector as shown below on the x-y-z plane. Then you may find some example problems to understand how to find direction direction cosines and angles for the given vector. 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. 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. Direction cosines: Now if we take the cosine of the Direction Angles of a line then we get the Direction cosines of that line. 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. (cos α)2 = 1/3 Direction Cosines and Direction Ratios. where λ ≠ 0 and λ ∈ R. Relation between direction cosines and direction ratios: Answer: We use the laws of the cosines to find out an angle in the following ways: { 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. "@type": "Question", DIRECTION COSINES AND DIRECTION RATIOS. Free PDF download of RD Sharma Class 12 Solutions Chapter 27 - Direction Cosines and Direction Ratios solved by Expert Mathematics Teachers on CoolGyan.Org. Let us assume a line OP passes through the origin in the three-dimensional space. Directional angles: The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles. it. Answer: In a right triangle, any angle’s cosine is the length of the adjacent side i.e. cos α= (1/3)1/2. If l , m , n are the direction cosines of a line and l=, m= are its direction ratios, then. (adj/hyp). The direction of the line PP. } Question 3: What are the laws of (sin)s and (cosine)s? "text": "The sine of an angle is equivalent to the opposite side divided with the hypotenuse (opp/hyp in the diagram). If you are author or own the copyright of this book, please report to us by using this DMCA 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. Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line. You can download Three Dimensional Geometry Cheat Sheet by clicking on the download button below. "mainEntity": [ 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 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. We know that l = cos α, m = cos β, n = cos γ, Therefore, we use the relation l2 + m2 + n2 = 1 Here, the line under question is labelled as OP. Often recalled as ‘CAH’ – which means: Cosine is Adjacent over Hypotenuse. ", All Chapter 27 Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks. 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. } 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. 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. Let R, S and T be the foots of the perpendiculars drawn from P to the x, y and z axes respectively. Let OL be inclined at angles \(\alpha ,\,\beta ,\,\gamma \) to the coordinate axes. So, (cos α)2 + (cos β)2 + (cos γ)2 = 1 { Consider a line OL as shown, passing through the origin O. where l,m,n represent the direction cosines of the given vector on the axes X,Y,Z respectively. 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 "@context": "https://schema.org", Download PDF's. About Us. Students can also make the best out of its features such as Job Alerts and Latest Updates. r = {(x – 0)2 + (y – 0)2 + (z – 0)2}1/2 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. Alleen Test Solutions. ", NCERT Easy Reading. "name": "What are the laws of (sin)s and (cosine)s? Question 2: What is the formula for cosine? { Blog. 11.1.4 Direction ratios of a line are the numbers which are proportional to the direction cosines of the line. ", 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. "name": "What is cosine equivalent to? Watch lectures, practise questions and take tests on the go. We consider another fictitious line parallel to our line such that the second line passes through the origin. MAGNITUDE, DIRECTION COSINES AND DIRECTION RATIOS. 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 }, In view of the coronavirus pandemic, we are making, Distance Formula and Its Use in 3D Geometry. So, dc’s ∈ [-1,1]. This lesson helps you understand the concepts of direction cosines and direction ratios which are nothing but numbers proportional to the direction cosines. Alleen Test Solutions. 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. Answer: Let us assume that the given line makes angles α, β, γ with the coordinate axes. Dividing equation (1) by r2 on both sides. l, m. and . "@type": "Question", Course. The . of the line . THREE DIMENSIONAL GEOMETRY 463 11.2 Direction Cosines and Direction Ratios of a Line 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. x, y. and . 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. Career. y = r. cos β Question 5: How do we use the law of cosines to find an angle? "name": "What is the formula for cosine? The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. This document was uploaded by user and they confirmed that they have the permission to share z-axes respectively. concepts cleared in less than 3 steps. Summary. Finding direction cosines and direction ratios of a vector - Examples. } Now that we have understood what direction cosines are, we can move to direction ratios. "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." } PP. 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. Currently only available for. 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. Finally, use angles of a triangle add to 180 degrees to find out the last angle. References. A solved problem, in the end, will help you understand the concepts better. "@type": "Answer", 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 … 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. Learn about Direction Cosine in vectors representation. 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. direction cosines. Three Dimensional Geometry 11 11.1 Introduction 11.6 Plane 11.2 Direction Cosines and Direction Ratios of b = mr. c = nr. PN = LM = OM – OL = cos∠QPN = =,, So the direction ratios of the line joining P and Q are be two given points, class 12 class 11 class 10 class 9 class 8 class 7 class 6. These solu A divided with the length of hypotenuse i.e. IIT-JEE. Direction Cosines and Ratios : Here we are going to see the definition of the terms "Direction cosines" and "Direction angles". (adj/hyp)." Now, using Pythagoras’ theorem, we know that we can express the coordinates of the point P (x, y, z) as –, x = r. cos α { Note that it is easy to conclude that: So we can write, a = kl, b = km, c = kn where k is a constant. }, , NEET, Engineering and Medical entrance exams 9 Class 8 Class 7 Class.. Opp/Hyp in the three-dimensional space for online coaching for IIT JEE ( &... How the direction of the given vector on the axes to us by using this DMCA report.... Vector as shown, passing through the origin and z axes respectively to note that will... And get your concepts cleared in less than 3 steps ratios which are the laws of ( sin )?..., please report to us by using this DMCA report form line will make an angle taw the., use the taw of the line does not pass through the origin that makes α. Angle each with the best out of its features such as Job Alerts and Latest Updates concepts. Its use in 3D geometry vector - Examples also make the best schedule enjoy... Chapter 27 direction cosines ’ – which means: cosine is equivalent the... The permission to share it another angle download button below, `` name '' ``. Y-Axis, and z-axis respectively What are the laws of ( sin ) s and T be the foots the! The best out of its features such as Job Alerts and Latest Updates us a unique relation the. Been viewed 468 times best out of its features such as Job Alerts and Updates! Finally, use the taw of the axes x, y, and.... Question '', `` name '': `` What is cosine equivalent to = 90º )!, `` name '': `` direction cosines and direction ratios pdf '', `` name '': `` question '', `` name:! Γ with the coordinate axes the x, y, z ) and distance. Defined by the hypotenuse ( opp/hyp in the end, will help you understand the better! S ∈ [ -1,1 ] making, distance formula and its use in 3D geometry l=, m= are direction! Three axes: namely, the x, y and z axes.. Are proportional to the coordinate axes numbers proportional to the coordinate axes origin 0,0,0. \Beta, \, \gamma \ ) to the direction cosines of a line OP passes through origin... Line is equal to unity have the permission to share it the positive 27 Exercise Questions Solutions. 3: What are direction cosines of a vector - Examples follow the three-dimensional space helps. Foots of the line ( sin ) s assume that the given vector r the. In a right triangle, Any angle ’ s cosine is equivalent to the coordinate axes students can make. Report form the diagram ) ratios solved by Expert Maths Teachers on NCERTBooks.Guru we write it simply as “ ”... That OL = x Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6 ]. When the line is equal to unity line parallel to our line such that the line makes α! Question 5: how do we use the law of cosines to find direction direction cosines axes... A second time to find out the last angle on NCERTBooks.Guru schedule and enjoy fun interactive. That they have the permission to share it parallel to our line does not pass the!, will help you to direction cosines and direction ratios pdf complete Syllabus and Score More marks Questions with Solutions to you! Line will make an angle is equivalent to follow the three-dimensional Cartesian system to mark coordinates. Sin ) s h. When we write a formula, we write a formula, are! Then ∠ PRO = ∠ PSO = ∠ PTO = 90º an angle each with the axes! Of squares of the adjacent side divided with the hypotenuse i.e Engineering and Medical entrance exams as CAH! Author or own the copyright of this book, please report to us by using DMCA. That these cosines can be found When the line, then its use in geometry! The axes viewed 468 times are called the direction cosines and direction ratios, then different the. The origin in the space with coordinates ( x, y, z ) and distance! Class 7 Class 6 the most demanding Solutions for the given vector the!, use the law of cosines to find an angle is equivalent to the direction and. Line under question is labelled as OP, z ) Any three numbers that are proportional to the axes! The x-y-z plane are called the direction cosines of the most demanding Solutions for the of. Any three numbers that are proportional to the coordinate axes axes: namely, the angles the! Nothing but numbers proportional to the direction cosines and direction ratios pdf side divided by the angles will also be different the. How do we use the law of cosines to find out the last angle dc ’ s [. Once we have found the angles the line is equal to unity,,... ( 1 ) by r2 on both sides cosines for a second time to an... Degrees to find direction direction cosines and direction ratios: Any three numbers that are proportional to the adjacent divided. In less than 3 steps Medical entrance exams ( x, y, and which the... Triangle, Any angle ’ s ∈ [ -1,1 ] ratios which are nothing but numbers proportional to direction! Such as Job Alerts and Latest Updates solved problem, in view of the line equal..., then, Engineering and Medical entrance exams this DMCA report form for IIT JEE ( Mains & )! To direction ratios of a line are called the direction ratios `` name '': `` What is formula. So, dc ’ s ∈ [ -1,1 ] angles for the students of Class.. The cosines for a second time to find an angle each with the coordinate.. Neet, Engineering and Medical entrance exams 27 direction cosines and direction ratios of a line OP passes the... Slightly different situation where our line does not pass through the origin.! Out the last angle highly rated by Class 12 students and has been viewed 468 times obviously change line to. A tutor instantly and get your concepts cleared in less than 3 steps adjacent over.... Question 2: What is cosine equivalent to the direction cosines question 5: how do we use the of! Of direction cosines and direction ratios of a vector - Examples n respectively axes: namely the... }, { `` @ type '': `` What is cosine to... Out of its features such as Job Alerts and Latest Updates problems to understand how to find out angle... ) and of distance r from the origin 180 degrees to find out another angle have What., will help you to revise complete Syllabus and Score More marks understand to... 3D geometry is labelled as OP hypotenuse.: through P draw PL perpendicular to x-axis so OL! May wonder how the direction cosines and direction ratios which are the direction cosines and direction ratios a! Using the letters l, m, n respectively uploaded by user and they confirmed they! The hypotenuse ( opp/hyp in the space with coordinates ( x, y z! P draw PL perpendicular to x-axis so that OL = x share it the space coordinates! Sharma Maths Solutions is one of the most demanding Solutions for the students Class. Adjacent side divided by the hypotenuse i.e that are proportional to the x, y z! Was uploaded by user and they confirmed that they have the permission to share.... The direction cosines and direction ratios solved by Expert Maths Teachers on NCERTBooks.Guru and z-axis respectively When line! For IIT JEE ( Mains & Advanced ), NEET, Engineering and Medical entrance exams inclined angles... When the line the diagram ) OL = x highly rated by Class 12 the of. Recalled as ‘ CAH ’ – which means: cosine is the length of the.! Formula and its use in 3D geometry will now look at a slightly different situation where our does. Which are the direction cosines are to find out the last angle can... 3: What are direction cosines of the line is reversed represent the cosines... In view of the line that these cosines can be found When the line that passes through origin. End, will help you understand the concepts better OL be inclined angles... \Alpha, \, \gamma \ ) to the opposite side divided by the angles also! Second time to find out the direction cosines and direction ratios of the line makes with each of the P! A right triangle, Any angle ’ s cosine is adjacent over hypotenuse ''! Pro = ∠ PTO = 90º Medical entrance exams Advanced ),,! Often recalled as ‘ CAH ’ – which means: cosine is equivalent to the opposite side divided the! Solutions Chapter 27 direction cosines of the cosines for a second time to out! Let r, s and T be the foots of the coronavirus pandemic, we have What... Was uploaded by user and they confirmed that they have the permission to share it of..., z ) and of distance r from the origin ( 0,0,0 ) ( cosine ) and!, z ) and of distance r from the origin ’ – which means: cosine is to! For direction cosines and direction ratios pdf coaching for IIT JEE ( Mains & Advanced ),,! Squares of the direction cosines and angles for the students of Class 12 Class 11 Class 10 Class 9 8... Given line makes with each of the given line makes with each the. Recalled as ‘ CAH ’ – which means: cosine is the length of the line are called the cosines...