The polar coordinate system uses distances and directions to specify the location of a point in the plane and is set up with a fixed point , the pole, and a ray from called the polar axis. Each point can be assigned polar coordinates where is the distance of and is the angle between the polar axis and . Any point can also be represented by
for any integer .
The connection between the two systems of polar and rectangular coordinates is clear where polar axis coincides with the positive axis. Using trigonometric ratios, we can change from polar to rectangular coordinates:
To change from rectangular to polar coordinates, use
These equations do not uniquely determine or . Make sure the values chosen for and give a point in the correct quadrant.
Similar to how a rectangular equation is an equation in and , a polar equation is an equation in and . The graph of a polar equation uses a grid consisting of circles centered at the pole and rays emanating from the pole.
In general, the graph of is a circle of radius centered at the origin because it consists of all points whose coordinate is , and we see that the equivalent equation in rectangular coordinates is after squaring both sides of the equation.
To sketch a graph of the polar equation , we first determine the polar coordinates of several points on the curve
The same points would be obtained if we allowed to range from to since is negative in that range. We plot these points, then join them to sketch the curve and the graph appears to be a circle. To express in rectangular coordinates:
In general, the graph of an equation of the form
is a circle with radius centered at the points with polar coordinates and , respectively.
Graphing Polar Equations
To graph some of the polar equations, instead of plotting points, we can first sketch the graph in rectangular coordinates for reference that enables reading at a glance the values of that correspond to a given . For example, the process of graphing is illustrated below.
Note the domain for in this case would be exactly : letting increase beyond or decrease beyond and we would be retracing the path. The heart-shaped curved is called a cardioid. In general, the graph of any equation of the form
is a cardioid.
The curve can be graphed in a similar manner:
This curve has four petals and is called a four-leaved rose. In general, the graph of an equation of the form
is an leaved rose if is odd or a leaved rose if is even.
Symmetry
The graph of a polar equation is symmetric:
with respect to the polar axis, if . Common triggers are equations involving only since .
with respect to the pole, if , or . Graph should be unchanged when rotated radians about the pole. Common triggers are certain equations like the one above or equations where is squared, for example, .
with respect to the line , if . Common triggers are equations involving only since .
In rectangular coordinates the zeros of the function correspond to the intercepts of the graph. In polar coordinates, the zeros of the function are the angles at which the curve crosses the pole. This is demonstrated in the following example of graphing :
This curve is called a limaçon (pronounced lih-muh-son, Middle French for snail). In general, the graph of an equation of the form
is a limaçon.
Consider the equation . We need to first determine the domain for , that is, finding the smallest interval that traces the entire curve without overlapping. The graph only repeats itself when the same value of is obtained at both and , thus we need to find the smallest integer that satisfies
For this equality to hold, must be a multiple of , therefore . In conclusion, we obtain the entire graph if we choose values between and .
Complex Numbers
We graph real numbers using the number line, which has one dimension. The fact that complex numbers have two components: a real part and an imaginary part, suggests we need two axes to graph complex numbers: the real axis and the imaginary axis. The plane determined by these two axes is called the complex plane. The complex number is represented by ordered pair in this plane. Similar to how the absolute value of a real number can be thought of as its distance from the origin, we define that the absolute value (or modulus) for complex number is
which is also the length of the line segment joining the origin to the point in the complex plane. If is an angle in standard position whose terminal side coincides with the line segment, we have
where and . is called the argument of . This is the polar form of the complex number in which the multiplication and division operations can be simplified:
which leads us to the De Moivre’s Theorem:
An th root of a complex number is a complex number such that . De Moivre’s Theorem suggests one th root of would be
The argument of can be replaced by for any integer , which would make the expression give a different value of for . Therefore, for any positive integer , complex number has distinct th roots
where the modulus of each th root is , the argument of the first root is . Repeatedly add to get the argument of each successive root. These observations show that the th roots of are spaced equally on the circle of radius when graphed.
Parametric Equations
Parametric Equations are a general method for describing any curve. Think of a curve as the path of a point moving in the plane; the and coordinates of the point are functions of time:
These equations are parametric equations for the plane curve that is the set of points , where , with parameter. A curve given by parametric equations can also be represented by a single rectangular equation with a process called eliminating the parameter. For example, consider the parametric equations
Solving for and substituting into the equation for , we get
Thus the curve is a parabola. Eliminating the parameter often helps us identify the shape of a curve. For another example:
Notice that , and since all points on the curve given by the parametric equations satisfy this equation, so the graph is a circle of radius centered at the origin. As increases from to , the point starts at and moves counterclockwise once around the circle.
We cover another example. As a circle of radius rolls along the axis, the curve traced out by a fixed point on the circumference is a cycloid for which we try to find the parametric equations. Let be the angle the circle has rolled through with the point started from the origin.
The distance that the circle has rolled must be the same as the length of the arc . From the figure we see
Vectors
A quantity determined completely by their magnitude, for example, mass, temperature and energy, is called a scalar. Quantities such as displacement, velocity and force that involve magnitude as well as direction are called directed quantities, which we represent through vectors. A vector in the plane is a line segment with an assigned direction, its length is called the magnitude. A vector denoted by (we often use boldface letter to denote vectors, so ) has initial point and terminal point .
To find the sum of any two vectors and , we sketch vectors equal to and with the initial point of one at the terminal point of the other, or if drawn starting at the same point, and would be the vector that is the diagonal of the parallelogram formed.
Multiplying a vector by a scalar has the effect of stretching or shrinking the vector. We define that the vector where is a real number has magnitude and has the same direction as if . The difference of two vectors and is defined by , as illustrated.
We can also describe vectors analytically by placing them in a coordinate plane, and represent them as ordered pairs of real numbers. Suppose we move units to the right and units upward to go from the initial point of the vector to the terminal point, then
where is the horizontal component of and is the vertical component of . It follows that if a vector is represented in the plane with initial point and terminal point , then
Note the vector is not the point . The vector itself represents only a magnitude and a direction, not a particular arrow in the plane, and therefore has many different representations depending on its initial point.
Two vectors are considered equal if they have equal magnitude and the same direction. In other words, two vectors are equal if and only if their corresponding components are equal, that is, for the vectors and , and .
The magnitude of a vector is
A unit vector is a vector of length . For instance, is a unit vector. Two special unit vectors are and defined by
We can express vectors in terms of them:
If we let be in the plane with its initial point at the origin, with a direction of , then
Thus we can also express as
We define the dot product of two vectors and to be
To get a dot product, the corresponding components are multiplied then added, resulting in a scalar instead of a new vector. Also notice the property
Let and be in the plane with their initial points at the origin, be the smaller of the angles formed by the two vectors, thus . Applying the Law of Cosines to the triangle formed by , and , and using the above property we get
Thus
where is the angle between the two nonzero vectors and . This is the Dot Product Theorem. Solving for , we get
which allows us to find the angle between two vectors by their components.
Two nonzero vectors and are called perpendicular if the angle between then is , and in that case,
Thus two nonzero vectors are perpendicular if and only if their dot product equals to .
We define the component ofalong (or the scalar projection ofonto) to be
where is the angle between and . Intuitively, the component of along is the magnitude of the portion of that points in the direction of . The below figure illustrates this concept.
In short, the component of along is
The vector parallel to and whose length is the component of along is the projection ofonto, given by
We often need to resolve a vector into the sum of two vectors, one parallel to and one orthogonal to in the form of , and in this case,
The below figure illustrates the idea.
For example, let and , then . If we resolve into and where is parallel to and is orthogonal to , then and .
