Vectors in Cartesian Cylindrical Spherical

[ha9981]

I do not understand when we are given a vector at point P(x,y,z) or in different forms cylindrical and spherical. What does it mean at point?? I mean aren't vectors supposed to start at origin, even if they don't how will that make a difference in their magnitude or angle between them.

For example we are given two vectors A(2,-2, 1) and B(1,2,-2). They are in different systems A is in spherical and B in in cylindrical and I don't have a problem handling that. Now what does it mean when it says: if they are perpendicular at spherical coordinate point P(2, theta, 60 deg) find theta. Now I know dot product is zero when they are perpendicular. I have no idea how how this has anything to do with theta as what does it mean when it says at point P?

 

[SammyS]

I do not understand when we are given a vector at point P(x,y,z) or in different forms cylindrical and spherical. What does it mean at point?? I mean aren't vectors supposed to start at origin, even if they don't how will that make a difference in their magnitude or angle between them.

For example we are given two vectors A(2,-2, 1) and B(1,2,-2). They are in different systems A is in spherical and B in in cylindrical and I don't have a problem handling that. Now what does it mean when it says: if they are perpendicular at spherical coordinate point P(2, theta, 60 deg) find theta. Now I know dot product is zero when they are perpendicular. I have no idea how how this has anything to do with theta as what does it mean when it says at point P?

Hello ha9981.

Position vectors start at the origin. In general vectors are associated with points quite removed from the origin. Perhaps it's the way that vector addition is presented that makes it seem that vectors may be moved around at will.

The direction of unit vectors, $$\hat{i},\,\hat{j},\,\hat{k}$$ for Cartesian coordinates, (x, y, z) is invariant under change of position.

The direction of the unit vectors for spherical coordinates, $$\hat{r},\,\hat{\theta},\,\hat{\phi}$$ depends on the coordinate of the point at which they are applied.

Translate your vectors, $$\vec{A},\,\text{ and }\vec{B}$$ into rectilinear coordinates at different positions, and you will get different results.

 

[SammyS]

Look at this link to http://mathworld.wolfram.com/SphericalCoordinates.html" .

At point (x, y, z) = (1, 0, 0):

$$\hat{r}=\hat{i},\ \hat{\theta}=\hat{j},\ \hat{\phi}=-\hat{k}\,.$$

At point (x, y, z) = (–1, 0, 0):

$$\hat{r}=-\hat{i},\ \hat{\theta}=\hat{j},\ \hat{\phi}=-\hat{k}\,.$$

 

[Fredrik]

I do not understand when we are given a vector at point P(x,y,z) or in different forms cylindrical and spherical. What does it mean at point?? I mean aren't vectors supposed to start at origin, even if they don't how will that make a difference in their magnitude or angle between them.

For example we are given two vectors A(2,-2, 1) and B(1,2,-2). They are in different systems A is in spherical and B in in cylindrical and I don't have a problem handling that. Now what does it mean when it says: if they are perpendicular at spherical coordinate point P(2, theta, 60 deg) find theta.

This doesn't make sense. Two vectors are either perpendicular or they're not. Coordinates don't have anything to do with it. However, if you're dealing with the tangent vectors of a curve, this is of course a bunch of different vectors, one at each point on the curve:

If C:[a,b]→ℝ³ is a curve, then C'(t)=(C'₁(t),C'₂(t),C'₃(t)) is the tangent vector of the curve at C(t). In particular, if C(t) is the position at time t, then C'(t) is the velocity at time t.

So is it possible that your book is talking about the tangent vectors of a curve?

Look at this link to http://mathworld.wolfram.com/SphericalCoordinates.html" .

At point (x, y, z) = (1, 0, 0):

$$\hat{r}=\hat{i},\ \hat{\theta}=\hat{j},\ \hat{\phi}=-\hat{k}\,.$$

At point (x, y, z) = (–1, 0, 0):

$$\hat{r}=-\hat{i},\ \hat{\theta}=\hat{j},\ \hat{\phi}=-\hat{k}\,.$$

These results are found by choosing a specific point in ℝ³ (i.e. a specific vector in ℝ³), and treating one of the three spherical coordinates as a variable while holding the others constant. This defines a curve through the point. The tangent of that curve is found by differentiation (as I described above), and the result is then normalized to length 1.

 

[blue_raver22]

I can understand your confusion about vectors in different coordinate systems. Let me try to explain it in a simple way.

Firstly, vectors can be defined at any point in space, not just at the origin. So when we say a vector at point P(x,y,z), it means that the vector starts at point P and extends in a certain direction and magnitude.

Now, the coordinate systems (Cartesian, cylindrical, and spherical) are just different ways of representing the same vector. In each system, the coordinates (x,y,z) or (r,θ,z) or (ρ,φ,θ) represent the position of the vector in that particular system.

In your example, vector A is given in spherical coordinates (ρ,φ,θ) and vector B is given in cylindrical coordinates (r,θ,z). This means that vector A starts at the origin and extends to point P(2,θ,60 degrees) in the spherical system, and vector B starts at the origin and extends to point (1,2,-2) in the cylindrical system.

Now, when we say that the vectors are perpendicular at point P, it means that the angle between them is 90 degrees when they are both extended to point P. This is where theta comes into play. We can use the dot product to find the angle between two vectors, and in this case, we know that the dot product is zero when the vectors are perpendicular. So by finding the value of theta, we can determine if the vectors are perpendicular at point P in the spherical system.

In summary, the coordinates (x,y,z) or (r,θ,z) or (ρ,φ,θ) simply represent the position of the vector in different systems, and the point P just specifies the location where the vectors are being compared. I hope this helps clarify the concept of vectors in different coordinate systems.