Question

Answer the following questions.

1.) Explain the significance of r<0 in an ordered pair (r, 0) in polar coordinates.
2.) Explain why a point in the plane can be represented by infinitely many ordered
pairs in polar coordinates.
3.) How is the component form of the vector related to the graph of the
vector in standard position?
4.)
A plane travels N30°W at 450 mph and encounters a wind blowing due west at 30 mph. (See Example 9)
a. Express the velocity of the plane Vp relative to the air in terms of I and J.
b. Express the velocity of the wind V in terms of i and j.
e. Express the true velocity of the plane vr in terms of i and j and find the true speed of the plane.

Answer 1

1. An ordered pair in polar coordinates (r, θ) represents a point in the plane with a distance of r from the origin and an angle of θ with respect to the positive x-axis. When r < 0, this means that the point is located on the opposite side of the origin from the positive x-axis. In other words, the angle θ is shifted by 180 degrees.

2. In polar coordinates, a point can be represented by infinitely many ordered pairs because the distance r and the angle θ are not unique. For example, the point (1, π/4) and the point (1, 9π/4) both represent the same point in the plane.

3. The component form of a vector (a, b) represents the horizontal and vertical displacement of the vector from its initial point to its terminal point. The graph of a vector in standard position is a line segment from the origin to its terminal point. The horizontal and vertical components of the vector correspond to the x- and y-coordinates of the terminal point, respectively.

4a. The velocity of the plane Vp relative to the air can be expressed as Vp = (450 cos(60°), 450 sin(60°)) - (30, 0) = (225 - 30, 389.71) = (195, 389.71) mph.

4b. The velocity of the wind V can be expressed as V = (-30, 0) mph.

4e. The true velocity of the plane vr can be expressed as vr = Vp + V = (195 - 30, 389.71) = (165, 389.71) mph. The true speed of the plane is the magnitude of the true velocity, which is approximately 414.7 mph.