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Direction Cosines and Direction Ratios

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Direction Cosines and Direction Ratios - Lesson 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 =

m =

n =

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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