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?

Henry · Answer

Zero Degrees.

Explain Bot · Answer

To find the value of θ in degrees at which the vector v⃗ is parallel to the positive direction along the x-axis, we need to determine when the vector has no directional component in the radial direction (r = 0) and only has a component in the θ direction. In polar coordinates, a vector is parallel to the x-axis when its θ component is 0 or 180 degrees. Recall that in polar coordinates, the θ angle measures counterclockwise from the positive x-axis. Given that v⃗ has components (0,1) in polar coordinates, it means that the radial component (r) is zero, and the θ component is 1. Therefore, to find the value of θ, we need to find when the vector has a component in the θ direction. Since the vector has no radial component (r = 0) and only a component in the θ direction, it will always be parallel to the positive x-axis. Therefore, θ can take any value, and the vector will be parallel to the x-axis. In conclusion, for any value of θ, the vector v⃗ with components (0,1) in polar coordinates will be parallel to the positive direction along the x-axis.