Notes On Direction Cosines and Direction Ratios - CBSE Class 12 Maths
Directional angles: The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles. Directional cosines: If Î±,ÃŸ and Î“ are the directional angles of a directed line L, then cos Î±, cos Î² and cos Î³ are called the directional cosines of directed line L. By convention: cos Î± = l cos Î² = m cos Î³ = n Thus, the directional cosines of l are: cos (Ï€-Î±) = - cos Î± cos (Ï€-Î²) = - cos ÃŸ cos (Ï€-Î³) = - cos Î“ If L1âˆ¥ L2: Direction cosines of L1 = Direction cosines of L2 Relation between direction cosines of a line: Let P = (x,y,z) OP = r â‡’ âˆš((x - 0)2 + (y - 0)2 + (z - 0)2) = r â‡’ âˆš(x2 + y2 + z2) = r â‡’ x2 + y2 + z2 = r2 ...... (1) âˆ´ OA = x OB = y OC = z Let Î±,ÃŸ and Î“ be the angles made by line L1 with the X, Y and Z axes, respectively. âˆ´ Direction cosines of line L are cos Î± ,cos Î² , cos Î³ In right-angled âˆ†OAP: cos Î± = l = OA/OP = x/r â‡’ x = lr ...(2) In right-angled âˆ†OBP: cos Î² = m = OB/OP = y/r â‡’ y = mr ...(3) In right-angled âˆ†OCP: cos Î³= n = OC/OP = z/r â‡’ z = nr ...(4) From equations (2), (3) and (4): x2 + y2 + z2 = l2 r2+ m2 r2 + n2 r2 â‡’ x2 + y2 + z2 = r2(l2 + m2 + n2) â‡’ l2 + m2 + n2 = (x2 + y2 + z2)/r2 â€¦(5) From equations (1) and (5): l2 + m2 + n2 = (x2 + y2 + z2)/(âˆš(x2 + y2 + z2))2 â‡’ l2 + m2 + n2 = (x2 + y2 + z2)/(x2 + y2 + z2) â‡’ l2 + m2 + n2 = 1 Direction Ratios x = lr y = mr z = nr Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line. If l,m,n are the direction cosines of a line, then a,b,c are its direction ratios such that: l = aÎ» m = bÎ» n = cÎ» where Î» â‰  0 and Î» âˆˆ R Relation between direction cosines and direction ratios: If l, m,n are the direction cosines of a line and a,b,c form a set of its direction ratios, then: l/a = m/b = n/c = k (constant) â‡’ l = ak ...(1)    m = bk ...(2)    n = ck ...(3) From equations (1), (2) and (3): l2 + m2 + n2 = (a2k2 + b2k2 + c2k2) â‡’ l2 + m2 + n2 = k2 (a2 + b2 + c2) â€¦ (4) We know that l2 + m2 + n2 = 1. Thus, in equation (4): k2 (a2 + b2 + c2) = 1 â‡’ k2 = 1/(a2 + b2 + c2) â‡’ k = Â± 1/âˆš(a2 + b2 + c2) â€¦ (5) From equation (1), (2), (3) and (5): l = a (Â± 1/âˆš(a2 + b2 + c2))= Â± a/âˆš(a2 + b2 + c2) m = b (Â± 1/âˆš(a2 + b2 + c2))= Â± b/âˆš(a2 + b2 + c2) n = c (Â± 1/âˆš(a2 + b2 + c2)) =Â± c/âˆš(a2 + b2 + c2) l = Â± a/âˆš(a2 + b2 + c2) m = Â± b/âˆš(a2 + b2 + c2) n = Â± c/âˆš(a2 + b2 + c2) Direction Cosines of a Line Passing through Two Given Points Direction cosines of a line passing through two given points: Let A= (x1,y1,z1) B = (x2,y2,z2) Direction cosines of given line = cos Î±, cos Î², cos Î³ In right-angled âˆ†BAC: cos Î² = AC/AB â€¦(1) AC = y2 - y1 AB = âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) â‡’ cos Î² = (y2 - y1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Similarly: cos Î± =  (x2 - x1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos Î³ = (z2 - z1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Direction cosines of a line passing through (x1,y1,z1) and (x2,y2,z3): cos Î± (l)   = (x2 - x1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos Î² (m) = (y2 - y1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos Î³ (n)  = (z2 - z1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Direction ratios of a line passing through (x1,y1,z1) and (x2,y2,z3): (x2 - x1), (y2 - y1), (z2 - z1) and (x1 - x2), (y1 - y2), (z1 - z2)

#### Summary

Directional angles: The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles. Directional cosines: If Î±,ÃŸ and Î“ are the directional angles of a directed line L, then cos Î±, cos Î² and cos Î³ are called the directional cosines of directed line L. By convention: cos Î± = l cos Î² = m cos Î³ = n Thus, the directional cosines of l are: cos (Ï€-Î±) = - cos Î± cos (Ï€-Î²) = - cos ÃŸ cos (Ï€-Î³) = - cos Î“ If L1âˆ¥ L2: Direction cosines of L1 = Direction cosines of L2 Relation between direction cosines of a line: Let P = (x,y,z) OP = r â‡’ âˆš((x - 0)2 + (y - 0)2 + (z - 0)2) = r â‡’ âˆš(x2 + y2 + z2) = r â‡’ x2 + y2 + z2 = r2 ...... (1) âˆ´ OA = x OB = y OC = z Let Î±,ÃŸ and Î“ be the angles made by line L1 with the X, Y and Z axes, respectively. âˆ´ Direction cosines of line L are cos Î± ,cos Î² , cos Î³ In right-angled âˆ†OAP: cos Î± = l = OA/OP = x/r â‡’ x = lr ...(2) In right-angled âˆ†OBP: cos Î² = m = OB/OP = y/r â‡’ y = mr ...(3) In right-angled âˆ†OCP: cos Î³= n = OC/OP = z/r â‡’ z = nr ...(4) From equations (2), (3) and (4): x2 + y2 + z2 = l2 r2+ m2 r2 + n2 r2 â‡’ x2 + y2 + z2 = r2(l2 + m2 + n2) â‡’ l2 + m2 + n2 = (x2 + y2 + z2)/r2 â€¦(5) From equations (1) and (5): l2 + m2 + n2 = (x2 + y2 + z2)/(âˆš(x2 + y2 + z2))2 â‡’ l2 + m2 + n2 = (x2 + y2 + z2)/(x2 + y2 + z2) â‡’ l2 + m2 + n2 = 1 Direction Ratios x = lr y = mr z = nr Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line. If l,m,n are the direction cosines of a line, then a,b,c are its direction ratios such that: l = aÎ» m = bÎ» n = cÎ» where Î» â‰  0 and Î» âˆˆ R Relation between direction cosines and direction ratios: If l, m,n are the direction cosines of a line and a,b,c form a set of its direction ratios, then: l/a = m/b = n/c = k (constant) â‡’ l = ak ...(1)    m = bk ...(2)    n = ck ...(3) From equations (1), (2) and (3): l2 + m2 + n2 = (a2k2 + b2k2 + c2k2) â‡’ l2 + m2 + n2 = k2 (a2 + b2 + c2) â€¦ (4) We know that l2 + m2 + n2 = 1. Thus, in equation (4): k2 (a2 + b2 + c2) = 1 â‡’ k2 = 1/(a2 + b2 + c2) â‡’ k = Â± 1/âˆš(a2 + b2 + c2) â€¦ (5) From equation (1), (2), (3) and (5): l = a (Â± 1/âˆš(a2 + b2 + c2))= Â± a/âˆš(a2 + b2 + c2) m = b (Â± 1/âˆš(a2 + b2 + c2))= Â± b/âˆš(a2 + b2 + c2) n = c (Â± 1/âˆš(a2 + b2 + c2)) =Â± c/âˆš(a2 + b2 + c2) l = Â± a/âˆš(a2 + b2 + c2) m = Â± b/âˆš(a2 + b2 + c2) n = Â± c/âˆš(a2 + b2 + c2) Direction Cosines of a Line Passing through Two Given Points Direction cosines of a line passing through two given points: Let A= (x1,y1,z1) B = (x2,y2,z2) Direction cosines of given line = cos Î±, cos Î², cos Î³ In right-angled âˆ†BAC: cos Î² = AC/AB â€¦(1) AC = y2 - y1 AB = âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) â‡’ cos Î² = (y2 - y1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Similarly: cos Î± =  (x2 - x1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos Î³ = (z2 - z1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Direction cosines of a line passing through (x1,y1,z1) and (x2,y2,z3): cos Î± (l)   = (x2 - x1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos Î² (m) = (y2 - y1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos Î³ (n)  = (z2 - z1)/âˆš((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Direction ratios of a line passing through (x1,y1,z1) and (x2,y2,z3): (x2 - x1), (y2 - y1), (z2 - z1) and (x1 - x2), (y1 - y2), (z1 - z2)

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