Chapter 6: Complex Numbers, Polar Coordinates, and Parametric Equations
- Page ID
- 113920
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- 6.1: Polar Coordinates
- Cartesian coordinates of a point are often called 'rectangular' coordinates. In this section, we introduce a new system for assigning coordinates to points in the plane -- polar coordinates. We start with an origin point, called the pole, and a ray called the polar axis. We then locate a point PP using two coordinates, (r,θ), where r represents a directed distance from the pole.
- 6.2: Graphs of Polar Equations
- In this section, we discuss how to graph equations in polar coordinates on the rectangular coordinate plane.
- 6.3: Polar Form of Complex Numbers
- In this section, we return to our study of complex numbers. We associate each complex number z=a+biz=a+bi with the point (a,b)(a,b) on the coordinate plane. In this case, the xx -axis is relabeled as the real axis, which corresponds to the real number line as usual, and the yy -axis is relabeled as the imaginary axis, which is demarcated in increments of the imaginary unit ii . The plane determined by these two axes is called the complex plane.
- 6.4: Hooked on Conics Again
- In this section, we revisit our friends the Conic Sections which we began studying in Chapter 7. Our first task is to formalize the notion of rotating axes. Armed with polar coordinates, we can generalize the process of rotating axes as shown below.
- 6.5: Vectors
- To answer questions that involve both a quantitative answer, or magnitude, along with a direction, we use the mathematical objects called vectors. The word 'vector' comes from the Latin vehere meaning 'to convey' or 'to carry.' A vector is represented geometrically as a directed line segment where the magnitude of the vector is taken to be the length of the line segment and the direction is made clear with the use of an arrow at one endpoint of the segment.
- 6.6: Parametric Equations
- There are scores of interesting curves which, when plotted in the xy-plane, neither represent y as a function of x nor x as a function of y. In this section, we present a new concept which allows us to use functions to study these kinds of curves.