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 z1 = r1(\,cos \theta1 + i \,sin \theta1) and z2 = r2(\,cos \theta2 + i \,sin \theta2) be two complex numbers. Then the following properties hold:

  • z1 z2 = r1 r2 (\,cos(\theta1 + \theta2) + i \,sin(\theta1 + \theta2))

  • \frac{z1}{z2} = \frac{r1}{r2} (\,cos(\theta1 - \theta2) + i \,sin(\theta1 - \theta2))

• Alternately,

  • (r1 \,cis \theta1)(r2 \,cis \theta2) = r1 r2 \,cis(\theta1 + \theta2)

  • \frac{r1 \,cis \theta1}{r2 \,cis \theta2} = \frac{r1}{r2} \,cis(\theta1 - \theta2)

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=\alpha1i+b1j = (\alpha1 ,b1) and w=\alpha2i+b2j = (\alpha2 ,b2) be vectors, and let a be a scalar. Then:
  \,\,\,+ \bar{w}=\,(\alpha1 + \alpha2) i +\, (b1 +b2) j = (a1 +a2), (b1 + b2)
  \,\,\,-\bar{w}= (\,\alpha1 - \alpha2) i + (b1 - b2) j = (a1 - a2 );(b1 -b2)
  \,\,\,\alpha v= (\alpha a1) i+ (\ alpha b1)j= (\, \alpha a1), (\alpha b1)
  $||||||= \sqrt{\alpha1^2 + b1^2}

• 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