If I differentiate that then, I getvela said:
What happens if you take the dot product of that with ##\hat r##?Istiakshovon said:If I differentiate that then, I get
$$\frac{\partial \vec r }{\partial \phi}=r (\cos \phi \hat j - \sin \phi \hat i)$$
But, I was thinking what I should do with the equation.
Show that vector is orthogonal to ##\vec r##Istiakshovon said:If I differentiate that then, I get
$$\frac{\partial \vec r }{\partial \phi}=r (\cos \phi \hat j - \sin \phi \hat i)$$
But, I was thinking what I should do with the equation.
$$\hat r \cdot \frac{\partial \vec r }{\partial \phi} = \hat r \frac{\partial \vec r }{\partial \phi} \cos \theta$$haruspex said:
How? :thinking Did you mean to graph?PeroK said:
You could use the dot product - correctly, of course.Istiakshovon said:
That's an original approach to say the least!Istiakshovon said:
To me $$\hat r r$$ represents the direction of phi is toward r. But, I think $$(\cos \phi \hat j - \sin \phi \hat i) \cos \theta$$ this are representing direction of phi is perpendicular. But, I am saying that by looking at original picture. Without that, I can't say that. So, how can I see that that's really perpendicular.PeroK said:
Note that it's either ##\phi## or ##\theta##, not both. Also, look up the dot product in Cartesian coordinates and note that:Istiakshovon said:To me $$\hat r r$$ represents the direction of phi is toward r. But, I think $$(\cos \phi \hat j - \sin \phi \hat i) \cos \theta$$ this are representing direction of phi is perpendicular. But, I am saying that by looking at original picture. Without that, I can't say that. So, how can I see that that's really perpendicular.
Andvela said:
Istiakshovon said:
Unit vectors are used to describe the direction of a vector in a coordinate system. They have a magnitude of 1 and are used to represent the direction of a vector without changing its length. This makes it easier to analyze and understand the direction of a vector in a given system.
The direction of a vector can be calculated by taking the dot product of the vector with the unit vectors in the x, y, and z directions. This will give you the components of the vector in each direction, which can then be used to determine the direction of the vector in the given coordinate system.
R roof and phi roof are unit vectors used in polar coordinates. R roof represents the direction of the radius vector, while phi roof represents the direction of the angular displacement from the positive x-axis. Together, they describe the direction of a vector in a polar coordinate system.
To convert from Cartesian coordinates (x, y, z) to polar coordinates (r, θ, φ), you can use the following equations: r = √(x^2 + y^2 + z^2), θ = arccos(z/r), and φ = arctan(y/x). These equations use the unit vectors r roof and phi roof to determine the direction of the vector in the polar coordinate system.
Yes, unit vectors can be used in three-dimensional space to describe the direction of a vector in a given coordinate system. In addition to the x, y, and z directions, a third unit vector, usually denoted as k cap, is used to represent the direction of the vector in the z direction. Together, these three unit vectors can describe the direction of a vector in three-dimensional space.