Study Notes: Calculus and Analytic Geometry – Polar Coordinates, Equations of Lines and Planes, and Distance in Space

Polar Coordinates and Graphs

Introduction to Polar Coordinates

Polar coordinates provide an alternative to Cartesian coordinates for representing points in the plane. Instead of using (x, y), a point is described by its distance from the origin r and the angle θ measured from the positive x-axis.

• Polar Coordinate: A point is represented as (r, θ).

• Conversion to Cartesian Coordinates:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

• Conversion from Cartesian to Polar:

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \arctan\left(\frac{y}{x}\right)$

Graphing Polar Equations

Polar equations describe curves using the variables r and θ. Common examples include circles, spirals, and roses.

• Example: The equation $r = 5 \cos \theta$ represents a circle centered at $(2.5, 0)$ with radius $2.5$.

• Graph Identification: To match a polar equation to its graph, analyze symmetry and intercepts.

Applications

• Polar coordinates are useful in problems involving circular symmetry, such as physics and engineering.

Equations of Lines and Planes in Space

Vector and Parametric Equations of a Line

In three-dimensional space, a line can be described using vector or parametric equations.

• Vector Equation: $\vec{r} = \vec{r}_0 + t\vec{v}$, where $\vec{r}_0$ is a point on the line and $\vec{v}$ is the direction vector.

• Parametric Equations:

  • $x = x_0 + at$

  • $y = y_0 + bt$

  • $z = z_0 + ct$

  • where $(x_0, y_0, z_0)$ is a point on the line and $(a, b, c)$ are the components of the direction vector.

• Example: The line through $(1, 2, 3)$ in the direction of $\langle 4, 5, 6 \rangle$ is $x = 1 + 4t$, $y = 2 + 5t$, $z = 3 + 6t$.

Equation of a Plane

A plane in space can be described by a point and a normal vector.

• General Equation: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$

• Standard Form: $ax + by + cz = d$

• Normal Vector: The vector $\langle a, b, c \rangle$ is perpendicular to the plane.

• Example: The plane through $(1, 2, 3)$ with normal $\langle 4, 5, 6 \rangle$ is $4(x-1) + 5(y-2) + 6(z-3) = 0$.

Intersection of Lines and Planes

To find the intersection, substitute the parametric equations of the line into the plane equation and solve for the parameter.

• Step 1: Write the parametric equations for the line.

• Step 2: Substitute into the plane equation.

• Step 3: Solve for $t$ to find the intersection point.

Distance in Space

Distance from a Point to a Plane

The shortest distance from a point to a plane is measured along the perpendicular from the point to the plane.

• Formula: For point $(x_1, y_1, z_1)$ and plane $ax + by + cz + d = 0$:

$\text{Distance} = \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$

• Example: Find the distance from $(2, 3, 4)$ to the plane $2x + 3y + 4z - 5 = 0$:

$\text{Distance} = \frac{|2 \cdot 2 + 3 \cdot 3 + 4 \cdot 4 - 5|}{\sqrt{2^2 + 3^2 + 4^2}} = \frac{|4 + 9 + 16 - 5|}{\sqrt{4 + 9 + 16}} = \frac{24}{\sqrt{29}}$

Solving Systems of Equations

Linear Systems

Systems of equations can be solved using substitution, elimination, or matrix methods.

• Example: Solve the system:

  • $x + y = 5$

  • $2x - y = 4$

  Solution: Add equations to eliminate $y$ and solve for $x$.

Table: Comparison of Coordinate Systems

System	Point Representation	Conversion to Cartesian	Applications
Cartesian	$(x, y)$	Direct	General geometry, algebra
Polar	$(r, \theta)$	$x = r \cos \theta$, $y = r \sin \theta$	Circular symmetry, physics

Additional info:

• Some questions involve matching graphs to equations, which is a common skill in calculus and analytic geometry.

• Problems on equations of lines and planes, and distances in space, are typical in multivariable calculus and analytic geometry courses.