DIRECTION COSINES AND RATIOS

Direction Cosines and Ratios :

Here we are going to see the definition of the terms "Direction cosines" and "Direction angles".

Then you may find some example problems to understand how to find direction direction cosines and angles for the given vector.

Let P be a point in the space with coordinates (x, y, z) and of distance r from the origin.

Let R, S and T be the foots of the perpendiculars drawn from P to the x, y and z axes respectively. Then

PRO  =  90°

PSO  =  90°

PTO  =  90°

OR = x, OS = y, OT = z and OP = r.

OR = x, OS = y, OT = z and OP = r.

(It may be difficult to visualize that PRO  =  PSO  =  PTO  =  90° in the figure; as they are foot of the perpendiculars to the axes from P; in a three dimensional model we can easily visualize the fact.)

Let α, β, γ be the angles made by the vector OP vector with the positive x, y and z axes respectively. That is,

PRO   =  α, POS   =  β, POT   =  γ

 In triangle OPR, ∠PRO = 90°, ∠POR =  α, OR = x, and OP = r. Hence cos α = OR/OP  =  x/r

In a similar way we can find that cos β  =  y/r and cos γ  =  z/r .

Direction angles :

Here the angles α, β, γ are called direction angles of the vector OP vector = r vector.

Direction cosine :

Cos α, cos β, cos γ are called direction cosines of the vector OP = xi vector + yj vector + zk vector.

Thus (x/r, y/r, z/r), where r  =  √(x2 + y2 + z2) are the direction cosines of the vector r vector  =  xi vector + yj vector + zk vector.

How to verify the direction ratios are direction cosines of a vector ?

Question 1 :

Verify whether the following ratios are direction cosines of some vector or not.

(i) 1/5 , 3/5 , 4/5

Solution :

In order to verify the given ratios are direction cosines, it has to satisfy the condition given below.

The sum of the squares of the direction cosines of r vector is 1.

That is ,

x2 + y2 + z =  1

r vector = xi vector + yj vector + zk vector

r vector = (1/5)i vector + (3/5)j vector + (4/5)k vector

x2 + y2 + z =  (1/5)2 + (3/5)2 + (4/5)2

=  (1/25) + (9/25) + (16/25)

=  (1 + 9 + 16)/25

=  26/25

x2 + y2 + z ≠ 1

Hence the direction ratios are not direction cosines of some vector.

(ii) 1/√2, 1/2 , 1/2

Solution :

r vector = (1/√2)i vector + (1/2)j vector + (1/2)k vector

x2 + y2 + z =  (1/√2)2 + (1/2)2 + (1/2)2

=  (1/2) + (1/4) + (1/4)

=  (2 + 1 + 1)/4

=  1

Hence the direction ratios are the direction cosines of some vector.

(iii) 4/3, 0, 3/4

Solution :

r vector = (4/3)i vector + (0)j vector + (3/4)k vector

x2 + y2 + z =  (4/3)2 + (0)2 + (3/4)2

=  (16/9) + 0 + (9/16)

=  (256 + 0 + 81)/144

x2 + y2 + z ≠ 1

Hence the direction ratios are not direction cosines of some vector.

After having gone through the stuff given above, we hope that the students would have understood, "Direction Cosines and Ratios".

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