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 the vector v⃗ is parallel to the positive direction along the x-axis, we need to convert the polar coordinates to Cartesian coordinates and analyze the x-component of the resultant vector.

In polar coordinates, (r, θ), the vector v⃗ can be expressed as v⃗ = (r*cos(θ), r*sin(θ)).

Given v⃗ = (0, 1) in polar coordinates, we have:
0 = r*cos(θ)
1 = r*sin(θ)

Since the x-component is zero, we know that r*cos(θ) = 0. This implies that θ should be either 90 degrees or 270 degrees, since cos(90) = 0 and cos(270) = 0.

To determine which value of θ is correct, we can analyze the y-component of the resultant vector. From the equation 1 = r*sin(θ), we can solve for r:
r = 1/sin(θ)

Since the y-component is 1 and sin(θ) cannot be zero, we know that r = 1. This means that the vector lies on the unit circle.

Considering the two possible values of θ, we can substitute them back into the polar coordinates expression:
For θ = 90 degrees:
v⃗ = (1*cos(90), 1*sin(90))
v⃗ = (0, 1)

For θ = 270 degrees:
v⃗ = (1*cos(270), 1*sin(270))
v⃗ = (0, -1)

As you can see, only when θ = 90 degrees, the vector v⃗ is parallel to the positive direction along the x-axis.

Therefore, the value of θ in degrees for which v⃗ is parallel to the positive direction along the x-axis is θ = 90 degrees.