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 α,ß and Γ are the directional angles of a directed line L, then cos α, cos β and cos γ are called the directional cosines of directed line L. By convention: cos α = l cos β = m cos γ = n Thus, the directional cosines of l are: cos (π-α) = - cos α cos (π-β) = - cos ß cos (π-γ) = - cos Γ If L1∥ L2: Direction cosines of L1 = Direction cosines of L2 Relation between direction cosines of a line: Let P = (x,y,z) OP = r ⇒ √((x - 0)2 + (y - 0)2 + (z - 0)2) = r ⇒ √(x2 + y2 + z2) = r ⇒ x2 + y2 + z2 = r2 ...... (1) ∴ OA = x OB = y OC = z Let α,ß and Γ be the angles made by line L1 with the X, Y and Z axes, respectively. ∴ Direction cosines of line L are cos α ,cos β , cos γ In right-angled ∆OAP: cos α = l = OA/OP = x/r ⇒ x = lr ...(2) In right-angled ∆OBP: cos β = m = OB/OP = y/r ⇒ y = mr ...(3) In right-angled ∆OCP: cos γ= n = OC/OP = z/r ⇒ z = nr ...(4) From equations (2), (3) and (4): x2 + y2 + z2 = l2 r2+ m2 r2 + n2 r2 ⇒ x2 + y2 + z2 = r2(l2 + m2 + n2) ⇒ l2 + m2 + n2 = (x2 + y2 + z2)/r2 …(5) From equations (1) and (5): l2 + m2 + n2 = (x2 + y2 + z2)/(√(x2 + y2 + z2))2 ⇒ l2 + m2 + n2 = (x2 + y2 + z2)/(x2 + y2 + z2) ⇒ 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 = aλ m = bλ n = cλ where λ ≠ 0 and λ ∈ 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) ⇒ l = ak ...(1)    m = bk ...(2)    n = ck ...(3) From equations (1), (2) and (3): l2 + m2 + n2 = (a2k2 + b2k2 + c2k2) ⇒ l2 + m2 + n2 = k2 (a2 + b2 + c2) … (4) We know that l2 + m2 + n2 = 1. Thus, in equation (4): k2 (a2 + b2 + c2) = 1 ⇒ k2 = 1/(a2 + b2 + c2) ⇒ k = ± 1/√(a2 + b2 + c2) … (5) From equation (1), (2), (3) and (5): l = a (± 1/√(a2 + b2 + c2))= ± a/√(a2 + b2 + c2) m = b (± 1/√(a2 + b2 + c2))= ± b/√(a2 + b2 + c2) n = c (± 1/√(a2 + b2 + c2)) =± c/√(a2 + b2 + c2) l = ± a/√(a2 + b2 + c2) m = ± b/√(a2 + b2 + c2) n = ± c/√(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 α, cos β, cos γ In right-angled ∆BAC: cos β = AC/AB …(1) AC = y2 - y1 AB = √((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) ⇒ cos β = (y2 - y1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Similarly: cos α =  (x2 - x1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos γ = (z2 - z1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Direction cosines of a line passing through (x1,y1,z1) and (x2,y2,z3): cos α (l)   = (x2 - x1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos β (m) = (y2 - y1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos γ (n)  = (z2 - z1)/√((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 α,ß and Γ are the directional angles of a directed line L, then cos α, cos β and cos γ are called the directional cosines of directed line L. By convention: cos α = l cos β = m cos γ = n Thus, the directional cosines of l are: cos (π-α) = - cos α cos (π-β) = - cos ß cos (π-γ) = - cos Γ If L1∥ L2: Direction cosines of L1 = Direction cosines of L2 Relation between direction cosines of a line: Let P = (x,y,z) OP = r ⇒ √((x - 0)2 + (y - 0)2 + (z - 0)2) = r ⇒ √(x2 + y2 + z2) = r ⇒ x2 + y2 + z2 = r2 ...... (1) ∴ OA = x OB = y OC = z Let α,ß and Γ be the angles made by line L1 with the X, Y and Z axes, respectively. ∴ Direction cosines of line L are cos α ,cos β , cos γ In right-angled ∆OAP: cos α = l = OA/OP = x/r ⇒ x = lr ...(2) In right-angled ∆OBP: cos β = m = OB/OP = y/r ⇒ y = mr ...(3) In right-angled ∆OCP: cos γ= n = OC/OP = z/r ⇒ z = nr ...(4) From equations (2), (3) and (4): x2 + y2 + z2 = l2 r2+ m2 r2 + n2 r2 ⇒ x2 + y2 + z2 = r2(l2 + m2 + n2) ⇒ l2 + m2 + n2 = (x2 + y2 + z2)/r2 …(5) From equations (1) and (5): l2 + m2 + n2 = (x2 + y2 + z2)/(√(x2 + y2 + z2))2 ⇒ l2 + m2 + n2 = (x2 + y2 + z2)/(x2 + y2 + z2) ⇒ 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 = aλ m = bλ n = cλ where λ ≠ 0 and λ ∈ 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) ⇒ l = ak ...(1)    m = bk ...(2)    n = ck ...(3) From equations (1), (2) and (3): l2 + m2 + n2 = (a2k2 + b2k2 + c2k2) ⇒ l2 + m2 + n2 = k2 (a2 + b2 + c2) … (4) We know that l2 + m2 + n2 = 1. Thus, in equation (4): k2 (a2 + b2 + c2) = 1 ⇒ k2 = 1/(a2 + b2 + c2) ⇒ k = ± 1/√(a2 + b2 + c2) … (5) From equation (1), (2), (3) and (5): l = a (± 1/√(a2 + b2 + c2))= ± a/√(a2 + b2 + c2) m = b (± 1/√(a2 + b2 + c2))= ± b/√(a2 + b2 + c2) n = c (± 1/√(a2 + b2 + c2)) =± c/√(a2 + b2 + c2) l = ± a/√(a2 + b2 + c2) m = ± b/√(a2 + b2 + c2) n = ± c/√(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 α, cos β, cos γ In right-angled ∆BAC: cos β = AC/AB …(1) AC = y2 - y1 AB = √((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) ⇒ cos β = (y2 - y1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Similarly: cos α =  (x2 - x1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos γ = (z2 - z1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Direction cosines of a line passing through (x1,y1,z1) and (x2,y2,z3): cos α (l)   = (x2 - x1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos β (m) = (y2 - y1)/√((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) cos γ (n)  = (z2 - z1)/√((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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