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?
Zero Degrees.
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 determine when v⃗ points in the same direction as the positive x-axis. In other words, we want to find the angle θ at which the vector has no radial component and points horizontally.
In polar coordinates, the vector v⃗ with components (0,1) means that the vector has a magnitude of 1 (r=1) and an angle of θ counterclockwise from the positive horizontal axis.
To find the value of θ, we can use the following steps:
1. Recall that the positive x-axis is at an angle of 0 degrees in the polar coordinate system.
2. Since the vector has no radial component (r=1), it means that the distance from the origin is fixed at a value of 1.
3. From the given components (0,1), we can see that the vector reaches a magnitude of 1 in the positive θ direction.
4. Therefore, the vector v⃗ is parallel to the positive x-axis when θ=0 degrees.
Hence, the value of θ in degrees at which v⃗ is parallel to the positive x-axis is θ=0 degrees.