Sum of angles with x,y and z axis made by a vector

In summary, the conversation discussed the relation between the angles of a vector with the three dimensional coordinate axes. It was determined that in 2D, the sum of the angles is always 180 degrees, however, in 3D this is not always the case. The question also asked about the relation between alpha and beta, but it was found that there is no direct relation between the angles. The conversation also mentioned using trigonometry to determine the angle that a vector makes with the z-axis. Finally, it was noted that for a vector to make equal angles with the three axes, the projections of the vector on each plane must make an angle of 45 degrees with both axes.
  • #1
rakesh
18
1
Homework Statement
sum of angles with x,y and z axis made by a vector
Relevant Equations
α + β + γ = 180°
I want to know if there is any proper relation between the angles of a vector with the three dimensional coordinate axes,

if the angles are ,α , β and γ,

will the sum of α, β and γ be 180 degress

that is α + β + γ = 180°,m finding the same to be true in a 2 D case where α + β = 90° and γ = , so the sum is 90°also will α + β
be always 90° as in 2D,

thanks.
 
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  • #2
A unit vector in spherical coordinates is written in the cartesian representation $$\mathbf{\hat r}=\sin\theta\cos\varphi ~\mathbf{\hat x}+\sin\theta\sin\varphi ~\mathbf{\hat y}+\cos\theta ~\mathbf{\hat z}$$The angles are defined as in the figure below. Do you know how to find the angle that an arbitrary vector makes with respect to the axes? If so, assume random numeric values for ##\theta## and ##\varphi##, calculate the angles this unit vector makes relative to the principal axes and see if they add up to 180°.

SphericalAngles.png
 
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  • #3
You might consider a vector that makes equal angles with the x, y, and z axes.
 
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  • #4
kuruman said:
A unit vector in spherical coordinates is written in the cartesian representation $$\mathbf{\hat r}=\sin\theta\cos\varphi ~\mathbf{\hat x}+\sin\theta\sin\varphi ~\mathbf{\hat y}+\cos\theta ~\mathbf{\hat z}$$The angles are defined as in the figure below. Do you know how to find the angle that an arbitrary vector makes with respect to the axes? If so, assume random numeric values for ##\theta## and ##\varphi##, calculate the angles this unit vector makes relative to the principal axes and see if they add up to 180°.

View attachment 322487
have tried to rotate the vector and get angles instead of using an equation but did not get much, will appreciate if u can give a clear and direct answer, thanks.
 
  • #5
TSny said:
You might consider a vector that makes equal angles with the x, y, and z axes.
i think each will be 60 degrees then but not sure.
 
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  • #6
rakesh said:
i think each will be 60 degrees then but not sure.
Try to work it out. Can you write down a vector in unit vector notation that would make equal angles to the three axes?
 
  • #7
TSny said:
Try to work it out. Can you write down a vector in unit vector notation that would make equal angles to the three axes?
my question is quite simple and i dnt want to enter into advanced approach of a unit vector and all.
 
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  • #8
rakesh said:
I want to know if there is any proper relation between the angles of a vector with the three dimensional coordinate axes,
α + β + γ = 180°
rakesh said:
if the angles are ,α , β and γ,

will the sum of α, β and γ be 180 degress
Yes

rakesh said:
my question is quite simple
Simple have been the responses from @kuruman and @TSny

Your question seems a little confusing to me.
 
  • #9
Lnewqban said:
α + β + γ = 180°

YesSimple have been the responses from @kuruman and @TSny

Your question seems a little confusing to me.
just see a 3d line and see the angles it makes with x, y and z axis, this is my question, is it confusing in any way.
 
  • #10
Consider a cube with edges of length 1 and let ##\mathbf A## be the vector shown.
1676745236093.png

Can you use trig to determine the angle that ##\mathbf A## makes to the z-axis?
 
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  • #11
sachin said:
just see a 3d line and see the angles it makes with x, y and z axis, this is my question, is it confusing in any way.
As you can see, it is perfectly possible for α + β + γ = 180°
Many combinations of those three angles can satisfy that equation.

For the particular case of each angle being 60 degrees, the projections of vector u on each plane x-y, y-z and x-z will make an angle of 45 degrees with both axes.

cosines.gif
 
  • #12
rakesh said:
my question is quite simple and i dnt want to enter into advanced approach of a unit vector and all.
The simple answer is "no". The angles don't always add to 180 degrees.
 
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  • #13
PeroK said:
The simple answer is "no". The angles don't always add to 180 degrees.
ok, so no relations between angles as we have seen in 2D, what about alpha plus bita, its also not equal to 90 ?
 
  • #14
rakesh said:
ok, so no relations between angles as we have seen in 2D, what about alpha plus bita, its also not equal to 90 ?
I thought you only wanted a simple answer?
 
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  • #15
Lnewqban said:
My concern is are these angles related in any way, m finding when they are equal the angles come to be cos inverse root 3 not 60 or 90 degrees.
PeroK said:
I thought you only wanted a simple answer?
Yes, as indirect answers will divert me from obvious vision and analysis on this simple case.
 
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  • #16
sachin said:
My concern is are these angles related in any way, m finding when they are equal the angles come to be cos inverse root 3 not 60 or 90 degrees.

Yes, as indirect answers will divert me from obvious vision and analysis on this simple case.
The key is that, using the inner product of unit vectors or otherwise we have:
$$\cos^2 X + \cos^2 Y + \cos^2 Z = 1$$Where ##X, Y, Z## are the angles between the vector and the coordinate axes. Note that if we take ##\cos Z = 0## to get the 2D case (x-y plane), then we can show that ##X + Y = \dfrac \pi 2## (or equivalent if we are outside the first quadrant).
 
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  • #17
sachin said:
My concern is are these angles related in any way,
Yes. For example, ##\cos^2 \alpha +\cos^2 \beta + \cos^2 \gamma = 1##. [Edit: @PeroK already posted this.]

sachin said:
m finding when they are equal the angles come to be cos inverse root 3 not 60 or 90 degrees.
Yes. Good.
 
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  • #18
Lnewqban said:
My concern is are these angles related in any way, m finding when they are equal the angles come to be cos inverse root 3 not 60 or 90 degrees.
PeroK said:
The key is that, using the inner product of unit vectors or otherwise we have:
$$\cos^2 X + \cos^2 Y + \cos^2 Z = 1$$Where ##X, Y, Z## are the angles between the vector and the coordinate axes. Note that if we take ##\cos Z = 0## to get the 2D case (x-y plane), then we can show that ##X + Y = \dfrac \pi 2## (or equivalent if we are outside the first quadrant).
any maximum or minimum values of the sum of the angles.
 
  • #19
sachin said:
any maximum or minimum values of the sum of the angles.
Sounds like a calculus problem with a constraint.
 
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  • #20
sachin said:
My concern is are these angles related in any way, m finding when they are equal the angles come to be cos inverse root 3 not 60 or 90 degrees.
Please, see excellent responses in posts #16 and #17 above.

Not only are the angles related, but the magnitude of the projection of the vector on each plane.

Retaking the case of 45 degrees projected over plane x-y:
Vector u will be projected at full length only if the vector is contained in plane x-y (γ = 90°).

Once the vector is tilting from that plane (γ < 90°, as shown in diagram of post #11), the length of its x-y projection becomes shorter and shorter until reaching a zero value when the vector becomes perpendicular to the x-y plane.

On the other hand, as long as the vector remains contained in plane x-y (γ = 90°), which is the 2-D particular case, its actual length will never change for any combination of α and β values (α + β = 90°).
 
  • #21
Lnewqban said:
Please, see excellent responses in posts #16 and #17 above.

Not only are the angles related, but the magnitude of the projection of the vector on each plane.

Retaking the case of 45 degrees projected over plane x-y:
Vector u will be projected at full length only if the vector is contained in plane x-y (γ = 90°).

Once the vector is tilting from that plane (γ < 90°, as shown in diagram of post #11), the length of its x-y projection becomes shorter and shorter until reaching a zero value when the vector becomes perpendicular to the x-y plane.

On the other hand, as long as the vector remains contained in plane x-y (γ = 90°), which is the 2-D particular case, its actual length will never change for any combination of α and β values (α + β = 90°).
if we will rotate the vector with a constant angle with x axis, will its angle with y axis go on changing ?
 
  • #22
Lnewqban said:
α + β + γ = 180°

YesSimple have been the responses from @kuruman and @TSny

Your question seems a little confusing to me.
as mentioned by others, i concluded its not 180, its changing, at some point of time it becomes 180 by chance.
 
  • #23
rakesh said:
if we will rotate the vector with a constant angle with x axis, will its angle with y axis go on changing ?
I am not sure that I fully understand your description.
If our vector shown in diagram of post #11 is made rotate around the x axis, keeping its tail at (0,0,0), it will describe the surface of a cone.

cone_iso1420774596274798465.jpg
 
  • #24
Lnewqban said:
α + β + γ = 180°

Yes
No.
Lnewqban said:
For the particular case of each angle being 60 degrees
… which cannot occur in Euclidean 3D space.

I'm confused by your responses. On the one hand you seem to say that the sum of the three angles will be 180°; on the other, you Liked post #12.

@rakesh/ @sachin (?), in N dimensions we have ##\Sigma_1^N\cos^2(\alpha_i)=1##.
For N=2, ##\cos^2(\alpha_1)+\cos^2(\alpha_2)=1##, which has the solution ##\alpha_1+\alpha_2=\pi/2##.
For a large number of dimensions, all of the angles can approach ##\pi/2## simultaneously, giving the approximation ##\Sigma_1^N(\pi/2-\alpha_i)^2=1##.
 
  • #25
Lnewqban said:
I am not sure that I fully understand your description.
If our vector shown in diagram of post #11 is made rotate around the x axis, keeping its tail at (0,0,0), it will describe the surface of a cone.

View attachment 322498
Yes, this is the exact case i meant, when alpha with x axis is kept constant, will beta be also constant
Lnewqban said:
I am not sure that I fully understand your description.
If our vector shown in diagram of post #11 is made rotate around the x axis, keeping its tail at (0,0,0), it will describe the surface of a cone.

View attachment 322498
Yes, this is the exact case i meant, when alpha with x axis is kept constant, will beta be also constant ? moreover i dnt understand what is "angle measured with positive y axis" is it when measured in anticlockwise direction ?
 
  • #26
I removed an off-topic post and replies to it.
 
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  • #27
Here's a contour plot for the value of ##\alpha+\beta+\gamma## as a function of the spherical coordinates ##\theta## and ##\phi## for the octant ##0 \leq \theta \leq 90^o## and ##0 \leq \phi \leq 90^o##. The numbers on the contours are the values of ##\alpha+\beta+\gamma##.

1676924687132.png


At any point on the square boundary of the plot, ##\alpha+\beta+\gamma = 180^o##. Everywhere else it is less than 180o. The minimum value of ##\alpha+\beta+\gamma## is approximately 164.2o and this occurs when the vector makes equal angles to the three axes: ##\cos \alpha = \cos \beta = \cos \gamma = \frac 1 {\sqrt 3}##; that is, when ##\theta = \cos^{-1}( \frac 1 {\sqrt 3}) \approx 54.7^o## and ##\phi = 45^o##.
 
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1. What does the sum of angles with x, y, and z axis made by a vector represent?

The sum of angles with x, y, and z axis made by a vector represents the direction of the vector in three-dimensional space. It shows the orientation of the vector with respect to the three coordinate axes.

2. How is the sum of angles with x, y, and z axis calculated?

The sum of angles with x, y, and z axis is calculated using trigonometric functions such as sine, cosine, and tangent. The angles are measured between the vector and each of the three axes.

3. Can the sum of angles with x, y, and z axis be negative?

Yes, the sum of angles with x, y, and z axis can be negative. This indicates that the vector is pointing in the opposite direction of the axis it is measured from.

4. Why is it important to know the sum of angles with x, y, and z axis of a vector?

Knowing the sum of angles with x, y, and z axis of a vector is important in understanding the orientation and direction of the vector in three-dimensional space. It can also be used in calculations and applications involving vectors, such as in physics and engineering.

5. How does the sum of angles with x, y, and z axis affect the magnitude of a vector?

The sum of angles with x, y, and z axis does not affect the magnitude of a vector. The magnitude of a vector is only determined by its length and is not affected by its direction or orientation in space.

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