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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 Î±,ß 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 ß

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±,ß 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 = ± 1/âˆš(a2 + b2 + c2) â€¦ (5)

From equation (1), (2), (3) and (5):

l = a (± 1/âˆš(a2 + b2 + c2))= ± a/âˆš(a2 + b2 + c2)

m = b (± 1/âˆš(a2 + b2 + c2))= ± b/âˆš(a2 + b2 + c2)

n = c (± 1/âˆš(a2 + b2 + c2)) =± c/âˆš(a2 + b2 + c2)

l = ± a/âˆš(a2 + b2 + c2)

m = ± b/âˆš(a2 + b2 + c2)

n = ± 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)