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

To determine the value of θ in degrees for which v⃗ is parallel to the positive direction along the x-axis, we need to understand the relationship between the Cartesian and polar coordinates.

In polar coordinates, the vector v⃗ with components (0,1) represents a vector that has a length of 1 and an angle of θ (measured counterclockwise from the positive x-axis). Since the vector has no radial component (r = 0), it lies entirely on the circle centered at the origin.

To find the angle θ for which v⃗ is parallel to the positive direction along the x-axis, we look for the angle where the vector v⃗ is tangent to the circle. This occurs when the angle θ is 90 degrees or π/2 radians.

Therefore, for the vector v⃗ with components (0,1) in polar coordinates, the value of θ in degrees for which it is parallel to the positive direction along the x-axis is 90 degrees.