What is the scalar product of the radial and the tangential unit

Question

What is the scalar product of the radial and the tangential unit

vectors (𝑟̂.𝑡̂), (the unit of measuring each of them is meter)?

Answer 1

As an AI language model, I am programmed to perform mathematical computations.

The scalar product (or dot product) of two vectors 𝑎 and 𝑏 is given by 𝑎 ⋅ 𝑏 = ∥𝑎∥ × ∥𝑏∥ × cos(𝜃), where ∥𝑎∥ and ∥𝑏∥ are the magnitudes of the vectors and 𝜃 is the angle between them.

The radial and tangential unit vectors are orthogonal (perpendicular) to each other, so their angle is 90 degrees and cos(90) = 0. Therefore, the scalar product of 𝑟̂ and 𝑡̂ is 0:

𝑟̂ ⋅ 𝑡̂ = ∥𝑟̂∥ × ∥𝑡̂∥ × cos(90) = 1 × 1 × 0 = 0 (in meters)

Answer 2

Well, the scalar product of the radial and tangential unit vectors (𝑟̂.𝑡̂) can be quite amusing! In this case, since the radial and tangential unit vectors are perpendicular to each other, their scalar product would be zero. So, it's almost like they're having a "zero-sum" game, where they cancel each other out. It's like asking two clowns to juggle and then expecting them not to drop anything!

Answer 3

To find the scalar product of the radial and tangential unit vectors, we need to determine the dot product between them.

The radial unit vector (𝑟̂) points in the direction of the radial line, which extends from the origin of a coordinate system to a point on a curve or surface. It is always perpendicular to the tangential unit vector.

The tangential unit vector (𝑡̂) points in the direction tangential to the curve or surface at a given point. It is always perpendicular to the radial unit vector.

Since the radial and tangential unit vectors are orthogonal (perpendicular) to each other, their dot product will be zero.

Therefore, the scalar product of the radial and tangential unit vectors (𝑟̂.𝑡̂) is zero.

Answer 4

To find the scalar product of the radial and tangential unit vectors (𝑟̂.𝑡̂), we need to multiply their respective components and then sum the results.

First, let's define the radial and tangential unit vectors:

- The radial unit vector (𝑟̂) points from the origin to a given point and is directed along the line connecting the origin to that point.
- The tangential unit vector (𝑡̂) points perpendicular to the radial vector and is tangent to the circular path at that point.

Now, since the scalar product is defined as the sum of the products of the corresponding components, we can express the radial and tangential unit vectors in terms of their components as follows:

- The radial unit vector 𝑟̂ can be expressed as 𝑟̂ = (𝑟_x, 𝑟_y, 𝑟_z), where 𝑟_x, 𝑟_y, and 𝑟_z are the components in the x, y, and z directions, respectively. In this case, as we are working in two dimensions (in the plane), we can ignore the 𝑟_z component, so 𝑟̂ becomes 𝑟̂ = (𝑟_x, 𝑟_y).
- The tangential unit vector 𝑡̂ can be expressed as 𝑡̂ = (𝑡_x, 𝑡_y, 𝑡_z), where 𝑡_x, 𝑡_y, and 𝑡_z are the components in the x, y, and z directions, respectively. Similarly, in two dimensions, we can ignore the 𝑡_z component, so 𝑡̂ becomes 𝑡̂ = (𝑡_x, 𝑡_y).

To find the scalar product (𝑟̂.𝑡̂), we multiply the corresponding components and sum the results:

𝑟̂.𝑡̂ = (𝑟_x, 𝑟_y)⋅(𝑡_x, 𝑡_y) = 𝑟_x*𝑡_x + 𝑟_y*𝑡_y

Therefore, the scalar product of the radial and tangential unit vectors is 𝑟_x*𝑡_x + 𝑟_y*𝑡_y.