Three Dimensional Geometry

Direction Cosines and Direction Ratios of a Line

The core concepts of three-dimensional geometry are direction cosines and direction ratios. What are direction cosines of a line that passes through the origin that makes angles with the coordinate axes? This lesson helps you understand the concepts of direction cosines and direction ratios which are nothing but numbers proportional to the direction cosines. A solved problem, in the end, will help you understand the concepts better.

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Direction Cosines

In three-dimensional geometry, we have three axes: namely, the x, y, and z-axis. Let us assume a line OP passes through the origin in the three-dimensional space. Then, the line will make an angle each with the x-axis, y-axis, and z-axis respectively.

The cosines of each of these angles that the line makes with the x-axis, y-axis, and z-axis respectively are called direction cosines of the line in three-dimensional geometry. Normally, it is tradition to denote these direction cosines using the letters l, m, n respectively.

Note that these cosines can be found only once we have found the angles that the line makes with each of the axes. Also, it is interesting to note that if we reverse the direction of this line, the angles will obviously change.

Consequently, the direction cosines i.e. the cosines of these angles will also be different once the direction of the line is reversed. We will now look at a slightly different situation where our line does not pass through the origin (0,0,0).

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Direction cosines when the line does not pass through the origin

You may wonder how the direction cosines are to be found when the line does not pass through the origin. The answer is simple. We consider another fictitious line parallel to our line such that the second line passes through the origin.

Now, the angles that this line makes with the three axes will be the same as that made by our original line and hence the direction cosines of the angles made by this fictitious line with the axes will be the same for our original line as well.



Here, the line under question is labelled as OP. It passes through the origin and we are to find out the direction cosines of the line. Note that we will follow the three-dimensional Cartesian system to mark the coordinates of the point P (x, y, z).

Let us assume that the magnitude of the vector is ‘r’ and the vector makes angles α, β, γ with the coordinate axes. Now, using Pythagoras’ theorem, we know that we can express the coordinates of the point P (x, y, z) as –

x = r. cos α
y = r. cos β
z = r. cos γ
r = {(x – 0)2 + (y – 0)2 + (z – 0)2}1/2
r = (x2 + y2 + z2)1/2

Now, as we stated earlier, we can replace cos α, cos β, cos γ with l, m, n respectively. Thus, we have –

x = lr
y = mr
z = nr

In the orthogonal system, we can represent r in its unit vector components form as –

\( r = x \hat{i} + y \hat{j}  + z \hat{k} \)

Using the relations we established above, we can substitute the values of x, y, z to get the following –

\(  \hat{r}  = lr\hat{i} + mr\hat{j}  + nr\hat{k} \)

So, \( \hat{r} \) = r/ | r | = l\(\hat{i} + m\hat{j} + n\hat{k} \)

By interpreting the above statement, it can be said that the direction cosines are the coefficients of the unit vectors \(\hat{i} , \hat{j} , \hat{k} \) when we express the unit vector \( \hat{r} \) in terms of its rectangular components.

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Direction Cosines and Direction Ratios of a Line

Direction Ratios

Now that we have understood what direction cosines are, we can move to direction ratios. Any numbers that are proportional to the direction cosines are called direction ratios, usually represented as a, b, c.

We have expressed earlier that r2 = (x2 + y2 + z2)……. (1)

In other words, OP2 = OA2 + OB2 + OC2

Dividing equation (1) by r2 on both sides,

1 = (x2 + y2 + z2)/ r2  = l2 + m2 + n2

This gives us a unique relation: the sum of squares of the direction cosines of a line is equal to unity. Note that it is easy to conclude that:

a ∝ l
b ∝ m
c ∝ n

So we can write, a = kl, b = km, c = kn where k is a constant.

Solved Example for You

Question 1: Find the direction cosines of the line that makes equal angles with each of the coordinate axes.

Answer: Let us assume that the given line makes angles α, β, γ with the coordinate axes. The direction cosines of the line are given by cos α, cos β, cos γ. We know that l = cos α, m = cos β, n = cos γ

Therefore, we use the relation l2 + m2 + n2 = 1
So, (cos α)2 + (cos β)2 ­­+ (cos γ)2 = 1
Since the line makes equal angles with the coordinate axes, cos α = cos β = cos γ

Thus, 3(cos α)2 = 1
(cos α)2 = 1/3
cos α= (1/3)1/2

Hence we can conclude that the line making equal angles with the coordinate axes has the direction cosines (1/3)1/2.

Question 2: What is the formula for cosine?

Answer: In a right triangle, any angle’s cosine is the length of the adjacent side i.e. A divided with the length of hypotenuse i.e. H. When we write a formula, we write it simply as “Cos”. Often recalled as ‘CAH’ – which means: Cosine is Adjacent over Hypotenuse.

Question 3: What are the laws of (sin)s and (cosine)s?

Answer: The Law of (Sin)s establish a relationship among the angles and the side lengths of triangle ABC: ‘a/sin(A)’ = ‘b/sin(B)’ = ‘c/sin(C)’. Sin is constantly positive in this range, whereas, the cosine is positive till 90 degrees where it turns out to be ‘0’ and is negative later.

Question 4: What is cosine equivalent to?

Answer: The sine of an angle is equivalent to the opposite side divided with the hypotenuse (opp/hyp in the diagram). The cosine is equivalent to the adjacent side divided by the hypotenuse i.e. (adj/hyp).

Question 5: How do we use the law of cosines to find an angle?

Answer: We use the laws of the cosines to find out an angle in the following ways:

  1. First of all, use the taw of the Cosines for calculating one of the angles.
  2. Then, use the law of the Cosines for a second time to find out another angle.
  3. Finally, use angles of a triangle add to 180 degrees to find out the last angle.

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