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 determine the value of θ in degrees at 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 right.
In polar coordinates, the vector v⃗ with components (0,1) represents a point on the unit circle (since r=1) that is 1 unit away from the origin and lies on the positive y-axis. This means that the angle θ represents the counterclockwise rotation from the positive x-axis to the vector v⃗.
Since we want to find the angle at which v⃗ is parallel to the positive x-axis, we need to find the angle at which v⃗ points directly right. In other words, we want θ to be 0 degrees or a multiple of 360 degrees. This is because a rotation of 0 degrees or a multiple of 360 degrees would mean no rotation at all, resulting in a vector parallel to the positive x-axis.
Therefore, the value of θ in degrees at which v⃗ is parallel to the positive x-axis is 0 degrees or any multiple of 360 degrees.