Direction Angles & Direction Cosines
Direction of a Vector in 3D
My previous post was about two primary ways in which we can describe the direction of a vector in two dimensional space. In this post, we will add in a third dimension(Z-axis) and discuss about a number of titles that are used to describe the direction of a vector in three dimensional space.
Let’s take an example of a vector in 3D in component form,
\(\vec{u} = 2\hat{i} + \frac{4}{\sqrt{2}} \hat{j} + 2\hat{k}\)
Just like for a 2D vector, the scalar components of a 3D vector can tell us the direction of the vector in the coordinate system. From the origin, we go 2 units in the positive X-direction; take a right angle turn and travel \(\frac{4}{\sqrt{2}}\) units(about 2.8 units) in the positive Y-direction; and finally travel 2 units in the positive Z-direction.
2 units in Positive X-direction
\(\frac{4}{\sqrt{2}}\) units in Positive Y-direction
2 units in Positive Z-direction
If you’d share this description with your friend, she will be easily able to locate the direction of the vector in 3D. The three scalar components when spoken only in context of the direction of a vector, they are referred to as Direction Numbers - the topic i covered in a recent post.
Direction by Direction Angles
The direction of a 3D vector can also be described by angles made by the vector with the coordinate axes. But here we need, not one not two but three angles formed with all three axes to describe the direction, unlike in 2D where only one angle with any one axis is needed. This one angle in 2D is coupled with the direction of the angle(clockwise or counterclockwise), and they together give us a complete description of the direction of the vector. For example, \(\vec{v}\) in 2D below is directed 45° Counterclockwise from the positive arm of X-axis.
The problem in 3D is that the direction of an angle(counterclockwise or clockwise) carries no meaning because the difference is not clear and obvious. To you the direction may be clockwise, but to someone sitting opposite to you, it is counterclockwise. Unlike in 2D, counterclockwise and clockwise are relative terms in 3D.
And so to kind of compensate for this “drawback”, we need, not one or two but three angles with all three axis to provide a proper and complete description of direction. Normally, those three angles of a vector are considered with positive arms of the three axes. For example, the direction of \(\vec{u}\) is 60° with positive arm of X-axis; 45° with positive arm of Y-axis; 60° with positive arm of Z-axis.
The three angles of a vector(or a line) with respect to the positive arms of X, Y and Z axes respectively, are called Direction Angles. Direction angles are denoted by Greek letters 𝛂, 𝛃 and 𝛄, pronounced as alpha, beta and gamma.
Just like the three scalar components of a vector, three direction angles of a vector give us a complete description of its direction. Also, note that unlike scalar components, direction angles will only talk about the direction aspect of the vector. To completely describe the vector both in magnitude and direction using direction angles, we need to specify both the magnitude and direction angles separately. For example, \(\vec{u}\) has magnitude 4 units and its direction angles are 60°, 45°, 60°.
One more way to describe the direction is by direction cosines.We will look at that in a moment but first i want to elaborate a bit further on direction angles.
\(\vec{u}\) makes 60° with positive X-axis, but we can also say that it makes 360° - 60° = 300° with positive X-axis.
Similarly, with Positive Y and Z axes, it makes 45° & 315° and 60° & 300° respectively. So to describe the direction of \(\vec{u}\), we can choose any one of the 2 angles with positive X-axis; any one of the 2 angles with positive Y-axis; and any one of the 2 angles with positive Z-axis. There are eight possible ways to select three angles from three pairs of angles. So a vector in 3D can have all the way upto 8 distinct sets of direction angles.
8 possible sets of Direction Angles of \(\vec{u}\) :
60°, 45°, 60° 60°, 45°, 300°
300°, 45°, 60° 300°, 45°, 300°
60°, 315°, 60° 60°, 315°, 300°
300°, 315°, 60° 300°, 315°, 300°
A vector does not have a unique set of direction angles. Any one of the sets described above can be used to describe the direction of \(\vec{u}\).
(Note : If a vector makes two 180° angles with the positive arm of any axis, then one set of da’s will be repeating, so there will be 7 distinct sets of da’s for such a vector.)
If you go to this page on Wikipedia, you will notice there is a restriction put on the values of direction angles. They are all defined in the range : 0° ≤ 𝛂, 𝛃 ,𝛄 ≤ 180°.
By restricting direction angles in the above mentioned range, we are giving them an element of uniqueness. Meaning, with this condition we are implying that a vector should have one and only one set of direction angles. For example, if we agree to this rule, the only set of direction angles of \(\vec{u}\) will be, 60°, 45°, 60°. We are discarding all other sets with one or more angles measuring >180°.
We obviously don’t have to go by this rule. But you will hardly ever come across a direction angle which is a reflex angle(>180). You will almost always encounter direction angles in problems in this range only. So don’t worry too much about it.
Direction Cosines of a Vector
Direction Cosines are simply the cosines of direction angles. Direction cosines of \(\vec{u}\) are,
cos 60°, cos 45°, cos 60°
\(\frac{1}{2},\hspace{1cm} \frac{1}{\sqrt{2}},\hspace{1cm} \frac{1}{2}\)
But our next question is interesting. Will the direction cosines of \(\vec{u}\) change for different sets of direction angles? How many sets of direction cosines can a vector have? 8?
Actually, the direction cosines of a vector is a UNIQUE set. In other words, a vector in 3D has one and only one set of direction cosines. The simple reason for this being the property which says that cosines of two explementary angles are equal.
cos(360° - θ) = cos(θ).
So that means,
cos(360 - 𝛂) = cos(𝛂) ; cos(360 - 𝛃) = cos(𝛃) ; cos(360 - 𝛄) = cos(𝛄)
The cosines of two explementary angles made by a vector with the positive arm of an axis are equal. So there is one and only one direction cosine of a vector associated w.r.t positive arm of each axis. We can check that the only set of dc’s of \(\vec{u}\) is \(\frac{1}{2} , \frac{1}{\sqrt{2}} , \frac{1}{2}\) by taking the cosines of angles in each set.
The direction cosines are also used as a description for the direction of a vector, in fact more so often than directions angles. Most textbooks give more importance to dc’s than da’s. Maybe that’s why they don’t go into too much detail about direction angles.
Direction Cosines often pop up in many different sections of the topic 3D geometry. There is also a very handy relation between three direction cosines of a vector and scalar components of that vector. This relation helps us in problem solving. I will talk about this relation in my future posts.
Direction Angles and Direction Cosines of a Line
Direction angles are also defined for a line in 3D. We can think of a line as two rays emerging from a common point, extending infinitely in two opposite directions.
Direction angles of a line are the three angles made by any one of the two rays with positive arms of the three respective axes. Just like a vector, a ray has upto 8 sets of direction angles. For a ray pointing in the opposite direction, it also has its own 8 sets of direction angles which are different from 8 sets of its opposite companion. So, a line in 3D has upto 8 + 8 = 16 total sets of direction angles! Any one of the 16 sets can be used describe the direction of that line.
(Note : if any ray is in the opposite direction of a positive arm, it will have 7 sets of da’s as one set will be repeating)
What about direction cosines of a line? How many sets of direction cosines a line can have?
Like a vector, a ray pointing in a direction has a unique set of direction cosines. So each ray of a line have its own unique set of direction cosines. So a line has in total 1 + 1 = 2 sets of direction cosines, one in each direction.
For the next topic, we will strictly keep the da’s within the range 0° ≤ 𝛂, 𝛃 ,𝛄 ≤ 180° to make things look simple. So a line has two sets of da’s in two directions, as well as two sets of dc’s in two directions.
Relation between two sets of dc’s of a Line
There exists a simple relation between two sets of direction angles of two rays of a line. Let’s say there is a line L passing through the origin. Direction angles of the ray that sits above the XY-plane are 60°, 45°, 60° respectively.
So this ray is making 60° angle with the positive arm of X-axis. Also, the angle between this ray and the opposite pointing ray is 180°. What angle will the opposite ray make with the positive X-axis?
From one ray to opposite ray, the angle is 180°; from one of the rays to positive X-axis somewhere in between, the angle is 60°. So from positive X-axis to the opposite ray, the angle will be 180° - 60° = 120°.
We can see that in the figure above. This, by definition, is one of the three direction angles of the opposite ray, made with positive X-axis. Similarly, the other two da’s of the opposite ray will be 180° - 45° = 135° and 180° - 60° = 120° with positive Y-axis and positive Z-axis.
If we had known the set of direction angles of the oppose ray instead, we could have found da’s of the ray sitting above the XY-plane in a similar fashion, by subtracting the three angles from 180°.
So to cut it short, if 𝛂, 𝛃, 𝛄 are da’s of any of the two rays of a line, then
180° - 𝛂, 180° - 𝛃, 180° - 𝛄 will be the da’s of the opposite ray. This is the relation between two sets of da’s of a line.
Is there also a relation between direction cosines too?
If cos 𝛂, cos 𝛃, cos 𝛄 are unique direction cosines of any of the two rays, then cos (180° - 𝛂), cos(180° - 𝛃), cos(180° - 𝛄) will be the unique direction cosines of the opposite ray.
But, cos(180° - θ) = - cos(θ).
Hence, cos (180° - 𝛂) = - cos(𝛂) ; cos(180° - 𝛃) = - cos(𝛃) ; cos(180° - 𝛄) = - cos(𝛄).
So the two sets of direction cosines of a line are : cos 𝛂, cos 𝛃, cos 𝛄 & - cos 𝛂, - cos 𝛃, - cos 𝛄.
In many texts, you will also see this written as : ± cos 𝛂, ± cos 𝛃, ± cos 𝛄.
In the example of line L above, its two sets of dc’s are :
cos 60°, cos 45°, cos 60° & - cos 60°, - cos 45°, - cos 60°
\(\frac{1}{2},\hspace{1cm} \frac{1}{\sqrt{2}},\hspace{1cm} \frac{1}{2}\) & \(- \frac{1}{2},\hspace{1cm} - \frac{1}{\sqrt{2}},\hspace{1cm} - \frac{1}{2}\)
We must be careful here. ± Symbol before the three dc’s does not imply that one set of dc’s will be ALL positive values and the other ALL negative values. Although for line L this seems to be the case but in general, this will only be true when all three da’s in one of the two directions are acute, just like in our example.
But one two or all three direction angles can be obtuse in nature too. One or two da’s can also be either 0° or 90°. There can also be a direction angle equal to 180°. If it is an acute angle, cosine ratio is positive. For obtuse angles and 180°, cosine ratio is negative. Cosine of 0° and 90° is zero.
Depending on the type of each direction angle, we can have a set of dc’s with a mix of positive and negative values and zero, for example, \(\frac{1}{2} , - \frac{1}{\sqrt{2}} , \frac{1}{2}\) .
The other set of dc’s in the opposite direction will simply be the negative of these values, \(- \frac{1}{2} , \frac{1}{\sqrt{2}} , - \frac{1}{2}\). And this is the simple relation between two sets of dc’s of a line.
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