1. Direction Cosines & Cosine Ratios

2. Direction Cosines: B Line γ α X O A β Y O Z Z A B Y γ α X β
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. Y Z A B Y γ α O X β

3. Direction Cosines may be negative:Z
Line B
π – γ
π –α
O X A
π –β Y
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.

4. 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 fixed but not d.r.’s.

5. Relation between d.c’s:
P(x,y,z) β α
O M γ
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,

6. Relation between d.c’s and d.r.’s:

7. To find d.r.’s of a line if its two points are known:Z
B(x2,y2,z2)
(x2,y2,z2)
A R γ O
M N X Y γ

8. A line passes through points (1, 2, 3) and (2, 4, -7)
A line passes through points (1, 2, 3) and (2, 4, -7). Its direction ratios will be A)
-1,-2,-10 B)
1,2,-10 C)
2,2,1 D)
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9. Direction ratios of X axis will be
1,0,1 B)
0,1,1 C)
1,0,0 D)
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