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?

Explain Bot · Answer

To determine the value of θ in degrees for which the vector v⃗ is parallel to the positive direction along the x-axis, we need to find the angle at which the vector points directly to the right and has no vertical component. In polar coordinates, the vector v⃗ with components (0,1) has a radial component of 0 and an angular component of 1. Since the radial component is 0, it means that the vector lies entirely on the circle centered at the origin. And since the angular component is 1, it means that the vector is tangent to the circle and points in the direction of increasing angle. To find the angle at which the vector points directly to the right, we need to find the angle θ when the vector is on the positive x-axis. In polar coordinates, the positive x-axis corresponds to an angle of 0 degrees or 360 degrees. So, the vector v⃗ is parallel to the positive direction along the x-axis when θ is either 0 or 360 degrees.