Cylindrical and Cartesian Coord. dot product

In summary, the problem is asking for the dot product between the unit vector in the x direction and the unit vector in the direction of θ in cylindrical coordinates. The equations provided show the conversion between Cartesian and cylindrical coordinates and the illustration helps visualize the unit vectors in cylindrical coordinates. To solve the problem, we first need to determine the components of the unit vector in the direction of θ.
  • #1
Cmwarre
1
0

Homework Statement



I'm given 2 unit vectors a_x and a_theta.
I need to find the dot product between the two.


Homework Equations



Conversion from Cylindrical to Cartesian

x = r * sin(theta)
y = r * cos(theta)
z = z

Conversion from Cartesian to Cylindrical

r = sqrt(x^2 + y^2)
theta = tan^-1(x/y)
z = z

The Attempt at a Solution



I'm just curious if you were my teacher what kind of answer would you want?
I think the point he's trying to prove is that theta is the angle between the x and y plane but I have no idea how to put that mathematically with the given formulas..
 
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  • #2
Cmwarre said:

Homework Statement



I'm given 2 unit vectors a_x and a_theta.
I need to find the dot product between the two.


Homework Equations



Conversion from Cylindrical to Cartesian

x = r * sin(theta)
y = r * cos(theta)
z = z

Conversion from Cartesian to Cylindrical

r = sqrt(x^2 + y^2)
theta = tan^-1(x/y)
z = z
A point P can be specified by its Cartesian coordinates (x, y, z) or by its cylindrical coordinates (r, θ, z). The equations above tell you how to convert one set of coordinates to the other.

This problem, however, is about something a bit different. There are unit vectors associated with every point. See, for instance, the illustration on this page. The unit vectors may depend on the coordinates (r, θ, z). The problem is asking you to find the dot product between what's labelled [tex]\hat{\theta}[/tex] on the illustration and the unit vector in the x direction.

A good place to start is figuring out or looking up what the components of aθ are.

The Attempt at a Solution



I'm just curious if you were my teacher what kind of answer would you want?
I think the point he's trying to prove is that theta is the angle between the x and y plane but I have no idea how to put that mathematically with the given formulas..
 

1. What is the difference between cylindrical and Cartesian coordinates?

Cylindrical coordinates use a combination of a distance from the origin, an angle from a fixed reference direction, and a height from a fixed plane to locate a point in space. Cartesian coordinates use a combination of three perpendicular axes (x, y, and z) to locate a point in space.

2. What is the dot product in cylindrical and Cartesian coordinates?

The dot product is a mathematical operation that takes two vectors and produces a scalar quantity. In cylindrical and Cartesian coordinates, the dot product is calculated by multiplying the corresponding components of the two vectors and then adding the products together.

3. How do you find the dot product in cylindrical coordinates?

In cylindrical coordinates, the dot product is calculated by multiplying the magnitude of one vector by the projection of the other vector onto it. This projection can be found by taking the cosine of the angle between the two vectors and multiplying it by the magnitude of the second vector.

4. What is the significance of the dot product in physics?

In physics, the dot product is used to calculate the work done by a force on an object, the angle between two vectors, and the projection of one vector onto another. It is also used to calculate the angle between a force and a displacement, which is important in determining the amount of energy transferred.

5. How do you use the dot product to find the angle between two vectors?

To find the angle between two vectors using the dot product, you can use the formula: cosθ = (A · B) / (|A| * |B|), where A and B are the two vectors and |A| and |B| are their magnitudes. Taking the inverse cosine of this value will give you the angle between the two vectors.

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