Polar Coordinates and Vectors

9.1 Polar Coordinates

• Introduction to polar $coordinate$ system as an alternative to the Cartesian coordinate system for plotting points in a plane.

• The Cartesian coordinate system labels a point as $(x, y)$, representing the signed distance from the y-axis and x-axis, respectively.

• In the polar coordinate system, the origin is called the pole, and points are described relative to the pole.

• The polar axis is the positive x-axis.

• A point in the polar coordinate system is represented by an ordered pair $(r, \theta)$.

  • If r > 0, then $r$ is the distance from the point to the pole; $\,theta$ is the angle formed by the polar axis and a ray from the pole through the point.

  • $(r, \theta)$ are called the polar coordinates of the point.

Polar Coordinates with Negative r

• If r < 0, the point $(r, \theta)$ is on the ray from the pole extending in the direction opposite the terminal side of $\,theta$ at a distance of $|r|$ units from the pole.

Non-Unique Coordinates in Polar System

• One drawback of the polar coordinate system is that points do not have unique coordinates.

• Any angle coterminal with $\,theta$, along with the same value for $r$, will yield a different set of coordinates that describes the same point.

• If r > 0, any angle coterminal with $\,frac{\,pi}{4}$ will work.

• If r < 0, any angle coterminal with $\frac{5\,pi}{4}$ will work.

• All polar coordinates that represent the same point as $(2, \frac{\,pi}{4})$ can be written as $(2, \frac{\,pi}{4} + 2\,pi k)$ or $(-2, \frac{5\,pi}{4} + 2\,pi k)$, where $k \in \mathbb{Z}$.

Conversion from Polar to Cartesian Coordinates

• To convert from polar to Cartesian coordinates, use the definitions of trigonometric functions: $\,cos \theta = \frac{x}{r}$ and $\,sin \theta = \frac{y}{r}$.

• Multiplying both equations by $r$ yields the conversion formulas: $x = r \,cos \theta$ and $y = r \,sin \theta$.

Conversion from Cartesian to Polar Coordinates

• To convert from Cartesian to polar coordinates:

  • Use $x^2 + y^2 = r^2$ to find $r$.

  • Use $tan \,theta = \frac{y}{x}$ to find $\,theta$, provided that $x \neq 0$.

Transforming Equations Between Coordinate Systems

• The equations used to convert points between systems can also transform equations.

• Two common techniques to transform from polar to rectangular form:

  • Multiply both sides of the equation by $r$.

  • Square both sides of the equation.

9.2 Polar Equations & Graphs

• To sketch polar graphs, use a polar grid consisting of concentric circles and lines.

• One way to sketch polar graphs is to transform the polar equation into a rectangular one.

• Let a ≠ 0. Then:

  • $r \,sin \theta = a$ is a horizontal line: a units above the pole if a > 0, a units below the pole if a < 0.

  • $r \,cos \theta = a$ is a vertical line: a units to the right of the pole if a > 0, a units to the left of the pole if a < 0.

• Let a > 0. Then:

  • $r = 2a \,sin \theta$ is a circle with radius a and center $(0, a)$.

  • $r = -2a \,sin \theta$ is a circle with radius a and center $(0, -a)$.

  • $r = 2a \,cos \theta$ is a circle with radius a and center $(a, 0)$.

  • $r = -2a \,cos \theta$ is a circle with radius a and center $(-a, 0)$.

• Checking for symmetry can reduce the number of points needed to sketch the graph.

• In polar coordinates:

  • Points $(r, \theta)$ and $(r, -\theta)$ are symmetric with respect to the polar axis (x-axis).

  • If $\,theta$ can be replaced with $-{\,theta}$ and the resulting equation is equivalent, the graph is symmetric with respect to the x-axis.

  • Points $(r, \theta)$ and $(r, \,pi - \theta)$ are symmetric with respect to the line $\theta = \frac{\,pi}{2}$ (the y-axis).

  • If $\,theta$ can be replaced with $\,pi - \theta$ and the resulting equation is equivalent, the graph is symmetric with respect to the y-axis.

  • Points $(r, \theta)$ and $(-r, \theta)$ are symmetric with respect to the pole (origin).

  • If $r$ can be replaced with $-r$ and the resulting equation is equivalent, the graph is symmetric with respect to the origin.

• A second method for sketching polar graphs is to plot points by plugging in values for $\,theta$ and finding the resulting value for $r$.

Cardioid

• A cardioid is a heart-shaped curve.

• Cardioids can be written in one of the following forms, where a > 0:

  • With y-axis symmetry: $r = a(1 + \,sin \theta)$ or $r = a(1 - \,sin \theta)$.

  • With x-axis symmetry: $r = a(1 + \,cos \theta)$ or $r = a(1 - \,cos \theta)$.

• All cardioids will pass through the pole.

Limaçon without a Loop

• Limaçons without loops can be written in one of the following forms, where a > b > 0:

  • With y-axis symmetry: $r = a + b \,sin \theta$ or $r = a - b \,sin \theta$.

  • With x-axis symmetry: $r = a + b \,cos \theta$ or $r = a - b \,cos \theta$.

• All limaçons without loops will not pass through the pole.

Limaçon with an Inner Loop

• Limaçons with inner loops can be written in one of the following forms, where b > a > 0:

  • With y-axis symmetry: $r = a + b \,sin \theta$ or $r = a - b \,sin \theta$.

  • With x-axis symmetry: $r = a + b \,cos \theta$ or $r = a - b \,cos \theta$.

• All limaçons with inner loops will pass through the pole twice.

• All cardioids and limaçons (with or without loops) have one of the following equations, where a and b are positive real numbers:

  • With y-axis symmetry: $r = a + b \,sin \theta$ or $r = a - b \,sin \theta$.

  • With x-axis symmetry: $r = a + b \,cos \theta$ or $r = a - b \,cos \theta$.

• The distinction between types comes from the relationship between a and b:

  • If a = b, the graph is a cardioid.

  • If a > b, the graph is a limaçon without a loop.

  • If a < b, the graph is a limaçon with an inner loop.

Rose Curves

• All roses can be written in one of the following forms, where $a \neq 0$ and n is an integer with $n \neq 0, \pm 1$:

  • $r = a \,cos(n \,theta)$

  • $r = a \,sin(n \,theta)$

  • If n is even, the graph has 2n petals. If n is odd, the graph has n petals.

Lemniscates

• All lemniscates can be written in one of the following forms, where $a \neq 0$:

  • $r^2 = a^2 \,sin(2 \,theta)$

  • $r^2 = a^2 \,cos(2 \,theta)$

• All lemniscates have graphs shaped like propellers.

Spirals

• Spirals can have a variety of equation types, so those are a little harder to classify.

9.3 Polar Form for Complex Numbers

• A complex number is a number in the form $a + bi$, where a and b are real numbers.

• Complex numbers can be plotted in the Argand plane (complex plane), where $a + bi$ is plotted in the same position as the point $(a, b)$ in the normal coordinate plane.

• Therefore, a complex number can have a polar form: If $z = a + bi$, then $z = (r \,cos \theta) + (r \,sin \theta)i = r(\,cos \theta + i \,sin \theta)$ is the polar form for z (a standard shorthand notation for the polar form is $z = r \,cis \theta$).

• In general, $r = \sqrt{a^2 + b^2}$, which is the same formula for the magnitude of z ($|z| = r$).

Operations with Complex Numbers in Polar Form

• Let $z<em>1 = r</em>1(\,cos \theta<em>1 + i \,sin \theta</em>1)$ and $z<em>2 = r</em>2(\,cos \theta<em>2 + i \,sin \theta</em>2)$ be two complex numbers. Then the following properties hold:

  • $z<em>1 z</em>2 = r<em>1 r</em>2 (\,cos(\theta<em>1 + \theta</em>2) + i \,sin(\theta<em>1 + \theta</em>2))$

  • $\frac{z<em>1}{z</em>2} = \frac{r<em>1}{r</em>2} (\,cos(\theta<em>1 - \theta</em>2) + i \,sin(\theta<em>1 - \theta</em>2))$

• Alternately,

  • $(r<em>1 \,cis \theta</em>1)(r<em>2 \,cis \theta</em>2) = r<em>1 r</em>2 \,cis(\theta<em>1 + \theta</em>2)$

  • $\frac{r<em>1 \,cis \theta</em>1}{r<em>2 \,cis \theta</em>2} = \frac{r<em>1}{r</em>2} \,cis(\theta<em>1 - \theta</em>2)$

DeMoivre's Theorem

• If $z = r(\,cos \theta + i \,sin \theta)$ is a complex number and n is a positive integer, then $z^n = r^n (\,cos(n \,theta) + i \,sin(n \,theta))$.

Finding Roots of Complex Numbers

• Let w and z be complex numbers, and suppose that $z^n = w$ for some positive integer n. In this case, we call z a complex nth root of w.

• When w = 1, the values of z are called the nth roots of unity.

• Let $w = r(\,cos \theta + i \,sin \theta)$ be a complex number, and let $n \geq 2$ be an integer. If $w \neq 0$, then there are n distinct complex nth roots of w, given by the formula

  • $z_k = \sqrt[n]{r} \,\,\,\,\,\,\,\,\,\,\, cis(\frac{\,theta + 2k\,pi}{n} )$, where $k = 0, 1, 2, …, n-1$.

9.4 - Vectors

• A vector is a quantity with both magnitude and direction.

• It is customary to write a vector as an arrow-the length of the arrow indicates the vector's magnitude while the arrowhead indicates the vector's direction.

• If P and Q are two distinct points in the coordinate plane, there is exactly one line through them. The points on the line joining P and Q is called the line segment PQ.

• If we order the points so that they proceed from P to Q, we have a directed line segment, or geometric vector, denoted as PQ. In this directed line segment, we call P the initial point and Q the terminal point.

• The magnitude of this vector is simply the distance between points P and Q.

• If a vector v has the same magnitude and direction as PQ, then we write v = PQ (some books use boldface type rather than write the arrows over the quantity in order to distinguish a vector from a number). We call these vectors equal.

• The zero vector, 0, is the vector whose magnitude is 0, and we generally don't assign it any direction.

• The sum v+w of two vectors is defined as follows: position the two vectors so that the terminal point of ✓ coincides with the initial point of w. The vector v+w is defined to be the vector whose initial point is the initial point of v and whose terminal point is the terminal point of w.

• Vector addition is commutative (u+v=V+u), and associative ($\bar{u}+ (\bar{v}+\bar{w}) = (\bar{u}+\bar{v})+\bar{w})$. The zero vector has the property that for any vector v,
  $\bar{v}+ 0 = 0 + \bar{v} = \bar{v}$''

• If v is a vector, then -v is a vector with the same magnitude as v but opposite direction.

• Clearly, based on this definition, v+(-v)=0.

• The difference v-w of two vectors is defined as follows: v-w=v+(-w).

• A scalar is a real quantity that only has magnitude. If a is a scalar and v is a vector, the scalar multiple av is defined as follows:

  • If a>0, av is the vector whose magnitude is a times the magnitude of v and whose direction is the same as v.

  • If a<0, av is the vector whose magnitude is a times the magnitude of v and whose direction is opposite that of v.

  • If α=0 or if v = 0, then av=0.

• Scalar multiples have the following properties, where v is a vector:

  • 1v=v

  • -1v =-V

  • $(\alpha + \beta)v= \alpha v + \beta v$

  • $\alpha (v+w)=\alpha v+\alpha w$

  • $\alpha (\beta v)=(\alpha \beta)v$

• We use the symbol $||$ to represent the magnitude of vector V, which will represent the length of v. Because of this, the following properties hold for V, where v is a vector and α is a scalar:

  • $||v||=0$ if and only if v = 0

  • $||\alpha v||=||\alpha| \,||v||$

• Further, if $||v||=1$, we call ū a unit vector.

• Algebraically, we represent a vector ✓ as ✓ = (a,b), where a and b are real numbers called the components of V. If the initial point of v = (a,b) is the origin, then we call v a position vector. If v = (a,b) is a position vector, then its terminal point is the ordered pair (a,b).

• Because any vectors with the same magnitude and direction are equal, we can see that if a vector v has initial point P₁ (✗₁,Y₁) and terminal point P2(X2,Y₂), then v=P₁₁₂ is equal to the position vector (×2 − × ₁, Y₂ — Y₁):

• Two position vectors are equal if and only if they have the same terminal point.

• Therefore, two vectors v and w are equal if and only if their corresponding components are equal; i.e., if v = (a₁,b₁) and w = (a₂, b₂), then v=w if and only if α₁ =α₂ and b₁ = b₂.

• An alternate algebraic notation for vectors, common in physical sciences, is as follows:

  • Let /= (1,0) be the unit vector whose direction is along the positive x-axis, and let j = (0,1) be the unit vector whose direction is along the positive y-axis. Any vector v = (a,b) may be written as v = (a,b) = a(1,0)+b(0,1)=ai+bj, and we call a and b the horizontal and vertical components of V, respectively. For example, for v = (5,4)=57+4j, 5 is the horizontal component while 4 is the vertical component.

• Let $v=\alpha<em>1i+b</em>1j = (\alpha<em>1 ,b</em>1)$ and $w=\alpha<em>2i+b</em>2j = (\alpha<em>2 ,b</em>2)$ be vectors, and let a be a scalar. Then:
  $\,\,\,+ \bar{w}=\,(\alpha<em>1 + \alpha</em>2) i +\, (b<em>1 +b</em>2) j = (a<em>1 +a</em>2), (b<em>1 + b</em>2)$
  $\,\,\,-\bar{w}= (\,\alpha<em>1 - \alpha</em>2) i + (b<em>1 - b</em>2) j = (a<em>1 - a</em>2 );(b<em>1 -b</em>2)$
  $\,\,\,\alpha v= (\alpha a<em>1) i+ (\ alpha b</em>1)j= (\, \alpha a<em>1), (\alpha b</em>1)$
  $||||||= \sqrt{\alpha1^2 + b1^2}$</p></li><li><p>For any nonzero vector ✓, the vector ū=$ v/||v|| is a unit vector in the same direction as v.
  same direction since u = (1/||V||) v
  is a
  positive scalar multiple
  magnitude, since ||u|| ||1/||V||| ||V|| = 1
  is a

• As a consequence of this, if u is a unit vector in the same direction as vector v, then
  v=||V||u.

• If a vector represents the speed and direction of an object, it is called a velocity vector. If a vector represents the direction and amount of force acting on an object, it is called a force vector. In many applications, a vector is described in terms of its magnitude and direction (ex. projectile launched at 25 miles per hour at 30° angle of inclination) rather than in terms of its components. Therefore, suppose we are given the magnitude || of a nonzero vector ✓ and the direction angle a, 0° ≤a<360°

• between v and i. Using standard position trigonometry and the definitions of sine and cosine in reference to the standard position angle on the unit circle, we can determine that ū=(cosa)+(sina)]. Along with the formula v = ||||u
  v=||((cosa)+(sina)])=(|||cosa)+(||sina)=(cosa,sina)

• Since forces can be represented by vectors, two forces