In three-dimensional geometry, lines that intersect the x, y, and z axes are referred to as the direction cosines of the line. They are defined as follows: To preserve consistency, the letters l, m, and n are often employed to denote these directional cosines.

Cosine angles may be determined only after the angle formed by the line with each of the triangle’s axes is established. It’s worth noticing that the angle changes when the line is reversed.

Introduction to direction cosine 

• Any vector that indicates its position concerning origin is called a position vector.
• Before talking about the directional cosines, consider a position vector.
• Consider three planes named x,y,z in the 3 d space with O as the origin, OQ is the reference line, and the length of the OQ line is q.
• Q is the vector in space with variable q.
• When a directed line OQ passes through the origin it makes α, β, and γ angles with the x, y, and z-axis, respectively, with O as the reference.
• α, β, and γ angles are called the direction angles of the line, and the cosine of these angles gives us the direction cosines.
• Suppose we extend the line OQ on 3 d places to know about the directional cosines so that we can take the supplement angle β as shown in the figure.
• The direction line gets reversed if we reverse the direction of the line OQ.
• After considering the angle made by the position vector and the direction of the line in the positive quarters, we can consider the position vectors of the line OQ.

If OQ = q, then we can see that

x= qcosα

y = qcosβ

z= qcosγ

Where k denotes the magnitude of the vector, and it is given by,

q = √(x–0)2+(y–0)2+(z–0)2

⇒q=x2+ y2+ z2

The cosines of direction angles cosα are represented by l

The cosines of direction angles cos β are represented by m

The cosines of direction angles cosγ are represented by n

x = qcosα =lq —————— (1)

y =qcosβ =mq————— (2)

z = qcosγ= nq————— (3)

We can also represent q in its unit vector components using the orthogonal system.

q = xi+ yj +zk

Substituting the values of x, y, and z, as lq, mq, nq respectively in the above equation

q= lq i + mq j+ nq z

⇒r = qq = l i + m j+ n z

• q – cosines of direction angles of a vector with a coefficient of the unit vector
• Unit vector q – is resolved by the rectangular components

The number that is proportional to the direction of cosine is called the direction ratio of the line. The direction ratio of the line is represented by a,b,c

Also OQ2 = OA2+OB2+OC2

q = x2+ y2 + z2

on dividing the equation q = x2+ y2 + z2   we get 

q2q2= x2q2+ y2q2+ z2q2

By using the equation 1,2,3

1 =xq2+ yq2+ zq2= l2 + m2 + n2

As mentioned above, the sum of the square of directional cosine = 1

Let a,b,c be the direction ratios of a line 

Where a∝l,b∝m, c∝n

so we get a=kl, b=km, c=kn

Here is the relation between direction ratios and direction cosines of a line is given by

la= mb= nc=k

We have already seen l2 + m2 + n2 = 1, from this we found k= 1a2+b2+c2

The value of k can be positive or negative based on the direction of the directed line

Solved Example

Consider the point P(x, y, z) having coordinates (1, 2, 3). Find the direction cosines of the line and the direct ratio with the origin point being O(0, 0, 0) for the given values in the question

Cos α,Cosβ,Cosγ= xr,yr, zr

 |r|= x2+ y2 + z2

|r|= 12+ 22 + 32  = √14

 We conclude Cos α,Cosβ,Cosγ(direction cosines)= 1√14,2√14, 3√14

The direction ratio of the given point P(x, y, z) will be 1:2:3.

Conclusion:

Direction cosine is a vector which represents the direction of line in a 3-D plane.If the angle line makes with x-axis= α  with y-axis is β  and with z-axis is γ then cos2α +cos2β +cos2γ=1.Direction ratio is just another way to represent the direction of a line.It’s advantage is we don’t have to deal with fractions for direction.