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Capsule – 2

(Direction Cosines & Cosine Ratios)

Slide No. Left Side Right Side

1 Direction Cosines:Let a line AB in space subtends

angles α, β, γ respectively with the positive direction of coordinate axes X, Y, Z.

We define cosα, cosβ, cosγ as direction cosines of the line AB.

Clearly, direction cosines fixes the direction cosines of a line in space. Clearly, parallel lines have same direction cosines.

d.c.’s are denoted by l, m, n respectively. i.e. l = cosα, m = cosβ and n = cosγ.

Direction of a line cannot be fixed in space by knowing any one or any two angles.

ZB

Lineγ

αX

A O β

YZ

γ A α B O X β

Y

2 Direction Cosines may be negative:

If a line in space makes angles α, β, γ respectively with the positive direction of coordinate axes X, Y, Z then consequently the line will make angles π - α, π - β, π – γ with negative direction of axes and hence d.c.’s may also be taken as cos π -α, cos π -β, cos π –γ.Thus, d.c.’s of the same line may also be taken as – cosα, –cosβ, –cosγ.

ZB

Line π – γ π –α

XA O

π –β

Y

3 Direction Ratios OR Direction Numbers:

Three numbers a, b, c proportional to d.c.’s l, m, n of a line in space are called direction ratios OR direction numbers of the line.

Clearly, d.c.’s of a line will be e-class/maths/nkm/kbagh

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fixed but not d.r.’s.

Slide No.

Left Side Right Side

4 Relation between d.c’s:Let d.c.’s of any line in space be l, m, n

and d.r.’s are a:b:c.Let P(x,y,z) be any point on the line and

PM is perpendicular from P on X axis.In right triangle PMO, P(x,y,z)

β α

O Mγ

5 Relation between d.c’s and d.r.’s:

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Slide No.

Left Side Right Side

6 To find d.r.’s of a line if its two pints are known: Z

B(x2,y2,z2) γ (x2,y2,z2) A R γ

O M N X

Y

Questions / Answers for Adobe Presenter:

Multiple Choice Questions:

Question (1): A line in space makes angles 300, 450 and 600 with positive direction of coordinate axes. Then its direction cosines are:

Question (2): A line passes through points (1, 2, 3) and (2, 4, -7). Its direction ratios will be

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Question (3): A line has direction ratios 1:2:3. Its direction cosines will be

Question (4): Direction ratios of X axis will be

Question (5): Direction ratios of the line inclined at equal angles with coordinate axes is

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