The most convenient way to express vectors in the two dimensional plane is in the familiar (x,y) Cartesian coordinates. However, one can express vectors in other coordinate systems as well. For example, another useful coordinate system for the plane is polar coordinates (r,θ), where r is the distance from the origin and θ is the angle counterclockwise from the positive horizontal axis. Consider the vector v⃗ with components (0,1) in polar coordinates. Unlike the (0,1) vector in Cartesian coordinates the direction of v⃗ changes depending on the angular coordinate of the point at which the vector is at. This is due to the fact that there is a 1 in the θ direction. Since the vector has no radial component, it always is tangent to the circle (points in the direction of increasing angle). For what value of θ in degrees is v⃗ parallel to the positive direction along the x-axis?

Question

The most convenient way to express vectors in the two dimensional plane is in the familiar (x,y) Cartesian coordinates. However, one can express vectors in other coordinate systems as well. For example, another useful coordinate system for the plane is polar coordinates (r,θ), where r is the distance from the origin and θ is the angle counterclockwise from the positive horizontal axis. Consider the vector v⃗ with components (0,1) in polar coordinates. Unlike the (0,1) vector in Cartesian coordinates the direction of v⃗ changes depending on the angular coordinate of the point at which the vector is at. This is due to the fact that there is a 1 in the θ direction. Since the vector has no radial component, it always is tangent to the circle (points in the direction of increasing angle). For what value of θ in degrees is v⃗ parallel to the positive direction along the x-axis?

Answer 1

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Answer 2

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Answer 3

To determine the value of θ in degrees for which the vector v⃗ is parallel to the positive direction along the x-axis, we can analyze the components in polar coordinates.

The vector v⃗ with components (r, θ) represents the vector from the origin to a point in the plane with distance r from the origin and an angle θ counterclockwise from the positive horizontal axis.

In this case, we are given that v⃗ has components (0, 1) in polar coordinates. The radial component (r) is 0, indicating that the vector points towards the origin. However, the angle component (θ) is 1, indicating that the vector is tangent to the circle and points in the direction of increasing angle.

To find the angle at which the vector v⃗ is parallel to the positive direction along the x-axis, we need to find the value of θ at which the vector points to the right horizontally.

In polar coordinates, the positive direction along the x-axis corresponds to an angle of θ = 0 degrees or θ = 360 degrees. Therefore, the value of θ for which v⃗ is parallel to the positive direction along the x-axis is θ = 0 degrees or θ = 360 degrees.