Direction Numbers of a Line in 3D

Direction Numbers(Direction Ratios) in 3D Geometry

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The term ‘Direction Numbers’ is generally introduced in 12th grade topic 3D geometry in CBSE syllabus. And here’s how it is defined. 

Any three numbers which are proportional to the direction cosines of a vector are called the direction numbers of the vector. 

That’s the definition of direction numbers that you will find in most indian standard textbooks, and in many online supplementary sources. 

This statement is followed by expressing the same thing in mathematical language. 

If a, b, c are direction numbers of a vector, then

a = s cos 𝜶, b = s cos 𝜷, c = s cos 𝜸 for any s ∈ R+

If from this you are able to understand what direction numbers are then great! But in my case it didn’t really sink in at first. And the worrying part was when I found that their usage comes up almost everywhere in the topic. It took some time for me to get comfortable with them. 

In this post I would like to present a different, perhaps an easier way to think about them, which also helped me understand what direction numbers are more clearly. Read on! 

To begin with, direction numbers can be of a vector or of a line in 3D.  

Direction Numbers of a vector 

Scalar components of a vector are direction numbers of that vector.

Think of any vector in 3D in component form. Let’s take a vector, 

$$\vec{u} = 2\hat{i} - 3\hat{j} + 4\hat{k}$$

2, -3 and 4 are scalar components of u(scalar components are coefficients of \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\)).  We can find both the magnitude as well as the direction of a vector in 3D from its three scalar components. To find the magnitude of \(\vec{u}\), just sum the squares of three numbers, and take the square root of what you get.

|\(\vec{u}\)| = \(\sqrt{2^2 + (-3)^2 + 4^2}\) = \(\sqrt{4 + 9 + 16}\) = \(\sqrt{29}\) units

How to find its direction?

From origin, go 2 units in the positive X-direction(2\(\hat{i}\)), followed by 3 units in the negative Y-direction(-3(\hat{j}\)), and finally 4 units in the positive Z-direction(4\(\hat{k}\)).

This will lead us to a point with coordinates (2, -3, 4). The directed line segment(arrow) from origin to (2, -3, 4) is a geometrical representation of \(\vec{u}\). The length of this arrow should be \(\sqrt{29}\) units. The direction in which the arrow is pointing is the direction of \(\vec{u}\) in 3D.  

So if we know the three scalar components of a vector, we can always locate the direction of the vector arrow in 3D. We can exactly pinpoint the direction of the vector. When scalar components of a vector are spoken in context of the direction of the vector they are referred to as its Direction Numbers. And that’s the first basic thing we ought to know about direction numbers. 

Generally, if a, b and c are scalar components of \(\vec{u}\) \(\longrightarrow\) a, b and c are Direction Numbers of \(\vec{u}\).

A vector has infinite number of direction numbers.

But now here’s a key part. The vectors’ own scalar components are not the only set of its direction numbers. The direction of a vector in 3D can also be described from scalar components of any other parallel vector. The scalar components of a parallel vector are direction numbers of the given vector as well! 

For example, suppose there is a vector, let’s call it \(\vec{p}\), in some 3 dimensional coordinate system. Suppose we don’t have any information about its magnitude or direction. But we do know that it is parallel to a vector \(\vec{v}\) whose scalar components are 1, 2 and 3.

$$\vec{v} = \hat{i} + 2\hat{j} + 3\hat{k}$$

Parallel vectors point in the same direction. So from the scalar components of parallel vector \(\vec{v}\) we can locate the direction of \(\vec{p}\) in 3D.

1, 2 and 3 are direction numbers of \(\vec{v}\) as well as of \(\vec{p}\). But it is important to note that scalar components of a parallel vector do not provide us with any information about the magnitude of \(\vec{p}\). We know where the arrow representing \(\vec{p}\) will be pointing but we don’t know what length it will have(that’s why i have drawn it dotted). 

The usage of the term ‘direction numbers’ is not done in context of the magnitude. The term Direction Numbers has a word ‘direction’ in it. It is important for us to understand that when we talk about direction numbers of any vector we are only concerned with the direction of that particular vector, that is where it points in 3D. 

In general, if we are told that a, b and c are direction numbers of any \(\vec{p}\), then it only means that there is some vector \(a\hat{i} + b\hat{j} + c\hat{k}\),

and the direction of this vector in 3D is the direction in which \(\vec{p}\) will be pointing. This vector could be \(\vec{p}\) itself or any other vector parallel to \(\vec{p}\), we don’t know and we don’t care. With direction numbers we are only concerned about the direction of that thing and nothing else!

So this brings us to two more crucial points regarding direction numbers of a vector. 

1. Scalar components of any parallel vector also qualify as the direction numbers of the given vector other than its own three scalar components. 

2. Direction numbers only point us to the direction of that vector in 3D

Ok. But now here’s another key part. In theory, we can define infinite number of vectors in any direction. You must have heard the statement, all parallel vectors are scalar multiples of each other. If \(\vec{v}\) is a vector pointing in some direction then 2\(\vec{v}\) is a vector twice the magnitude in the same direction; ¼ \(\vec{v}\) is a vector one-fourth the magnitude of \(\vec{v}\) pointing in the same direction; \(\sqrt{3} \vec{v}\) is \(\sqrt{3}\) times the magnitude of \(\vec{v}\) in the same direction.

In general, for all possible positive real values of ‘s’, s\(\vec{v}\) are all possible vectors pointing in the same direction as \(\vec{v}\).  

Also, since \(\vec{v}\) is parallel to \(\vec{p}\), every other vector in that direction is also parallel to \(\vec{p}\). So if we are talking about the direction numbers of \(\vec{p}\), there are infinitely many sets, -1, 2 and 3; -2, 4 and 6; \(\frac{-1}{4}, \frac{1}{2}, \frac{3}{4}\);\(-\sqrt{3}, 2\sqrt{3}, 3\sqrt{3}\), etc; they all can describe the direction of (\vec{p}\).

In general, if a, b and c are dn’s of \(\vec{p}\), then so are sa, sb and sc for all positive real values of s. 

If we know one set of direction numbers we can determine infinitely many other sets! 

But you’ll probably think, why do we want to find another set of dn’s when we already have one set?

Well that’s just our choice to work with any set. Sometimes in problems you will find an irrational number or some intricate fraction present as dn’s, which people don’t normally like working with. If it is the same irrational number or fraction present in all three, we can just multiply all three of them by a cleverly chosen number so that they change to integers or much something easier to work with!

Direction Numbers of a straight Line in 3D

Direction numbers are more affiliated to lines in 3D than to vectors. They are quite a useful tool that help in finding an equation of a straight line. Not only that, they are used pretty much in every concept in the topic that has to do with a line in 3D and its equation. 

A vector arrow in 3D is just some length in a particular direction. But a line is made up of infinitely many points. A line extends infinitely in two opposite directions, and all the points that make up the line have a fixed location in space.

Direction numbers of a line tell us the two directions in which the line extends in 3D. If we are given that a, b and c are direction numbers of a line then what it means is that one of the two directions of this line is same as the direction of the vector, \(a\hat{i} + b\hat{j} + c\hat{k}\) in 3D.

And obviously the line will also extend infinitely in the opposite direction as well. 

For example, let’s say there is a line L in 3D and -1, 2 and 3 are its direction numbers. 

Let \(\hat{v} = - \hat{i} + 2\hat{j} + 3\hat{k}\)

From where we are looking at the coordinate axes, this vector \(\hat{v}\) points somewhere between to our right, up and away from us. Line L will be stretching in the same direction as well as in the direction opposite to this. But can we really draw L in 3D from its dn’s? Where will the line L be in the 3D space?   

The problem here is that there can be infinitely many parallel lines extending in these two directions. Any of them can be L. Dn’s of a line will only locate us to the direction of the line!

To determine which line it is we need more information. If we know the coordinates of just one point on the line then along with direction numbers, we can clearly specify where the line is in 3D! Suppose we know that the line L passes through a point whose coordinates are (2, -1, 1). 

There can be one and only one line that passes through (2, -1, 1) and in a given direction -1, 2 and 3. With just any one set of direction numbers and coordinates of any one point on the line we can uniquely describe that line in 3D. In fact using these two parameters we can also find the Cartesian equation of the line - an algebraic relation satisfied by coordinates of all points on the line - something that we will see in a different post.  

Also, just like vectors, a line in 3D has infinitely many sets of direction numbers. If a, b and c are dn’s of a line then so are sa, sb and sc. But the difference here is that ‘s’ can be any non-negative real number(positive or negative, but not zero) unlike in the case of vectors where ‘s’ was strictly positive. 

Before I end this post, I just want to quickly talk about two things. Direction numbers have another name, direction ratios. I haven’t really used the second name throughout the post except in the title. In a different post I will talk about why there is ‘ratio’ in its name. 

Also, i had merely stated the definition in the beginning and didn’t really elaborate it at any point throughout the post. This definition will only make sense after we know what direction cosines are - the topic i wish to cover in upcoming posts.