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?

To find the value of θ in degrees for which the vector v⃗ is parallel to the positive direction along the x-axis, we need to convert the polar coordinates (r, θ) into Cartesian coordinates (x, y) and determine the θ value that makes the y-component (or the vertical component) of the vector equal to zero.

In polar coordinates, the vector v⃗ can be expressed as (r, θ) = (0, 1), where r represents the distance (magnitude) and θ represents the angle counterclockwise from the positive horizontal axis.

Since the vector has no radial component (r = 0), it implies that the vector points directly along the tangent of the circle at the origin, which means it aligns with the positive x-axis. For v⃗ to be parallel to the positive x-axis, the y-component must be zero.

To convert from polar coordinates to Cartesian coordinates, we use the following formulas:

x = r * cos(θ)
y = r * sin(θ)

In this case, since r = 0, the formulas simplify to:

x = 0 * cos(θ) = 0
y = 0 * sin(θ) = 0

Therefore, no matter what the value of θ is, the y-component will always be zero, making the vector v⃗ parallel to the positive direction along the x-axis.

In conclusion, for any value of θ, the vector v⃗ is parallel to the positive direction along the x-axis.