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

[rakesh]

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.

 

[kuruman]

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°.

 

[TSny]

You might consider a vector that makes equal angles with the x, y, and z axes.

 

[rakesh]

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.

 

[rakesh]

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.

 

[TSny]

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?

 

[rakesh]

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.

 

[Lnewqban]

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°

if the angles are ,α , β and γ,

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

Yes

my question is quite simple

Simple have been the responses from @kuruman and @TSny

Your question seems a little confusing to me.

 

[sachin]

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.

 

[TSny]

Consider a cube with edges of length 1 and let $\mathbf A$ be the vector shown.

Can you use trig to determine the angle that $\mathbf A$ makes to the z-axis?

 

[Lnewqban]

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.

 

[PeroK]

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.

 

[rakesh]

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 ?

 

[PeroK]

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?

 

[sachin]

View attachment 322491

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.

I thought you only wanted a simple answer?

Yes, as indirect answers will divert me from obvious vision and analysis on this simple case.

 

[PeroK]

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).

 

[TSny]

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.]

m finding when they are equal the angles come to be cos inverse root 3 not 60 or 90 degrees.

Yes. Good.

 

[sachin]

View attachment 322491

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.

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.

 

[PeroK]

any maximum or minimum values of the sum of the angles.

Sounds like a calculus problem with a constraint.

 

[Lnewqban]

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°).

 

[rakesh]

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 ?

 

[rakesh]

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.

 

[Lnewqban]

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.

 

[haruspex]

α + β + γ = 180°

Yes

No.

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 $\Sigma_1^N(\pi/2-\alpha_i)^2=1$.

 

[sachin]

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

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 ?

 

[mfb]

I removed an off-topic post and replies to it.

 

[TSny]

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 $\alpha+\beta+\gamma$.

At any point on the square boundary of the plot, $\alpha+\beta+\gamma = 180^o$. Everywhere else it is less than 180^o. The minimum value of $\alpha+\beta+\gamma$ is approximately 164.2^o 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 $\phi = 45^o$.