Essential Concepts

Parametric Equations

• Parametric equations are a set of equations where both [latex]x[/latex] and [latex]y[/latex] are expressed as functions of a third variable [latex]t[/latex] (the parameter, often representing time): [latex]x = f(t)[/latex] and [latex]y = g(t)[/latex]. The set of ordered pairs [latex](x(t), y(t))[/latex] forms a plane curve.

Creating Parametric Equations from Rectangular Form

Given a rectangular equation like [latex]y = x^2 - 1[/latex]:

• Choose a parameterization for [latex]x[/latex], typically [latex]x(t) = t[/latex] (simplest choice)
• Substitute this into the rectangular equation to get [latex]y(t)[/latex]
• Example: If [latex]x(t) = t[/latex], then [latex]y(t) = t^2 - 1[/latex]
• There are infinitely many ways to parameterize the same curve

Graphing Parametric Equations:

1. Create a table with columns for [latex]t[/latex], [latex]x(t)[/latex], and [latex]y(t)[/latex]
2. Choose values for [latex]t[/latex] in increasing order
3. Calculate corresponding [latex]x[/latex] and [latex]y[/latex] values
4. Plot the [latex](x, y)[/latex] points
5. Draw arrows showing the direction as [latex]t[/latex] increases (orientation)

Converting parametric equations to rectangular form:

For Polynomial, Exponential, or Logarithmic Equations:

1. Solve the simpler equation for [latex]t[/latex]
2. Substitute this expression for [latex]t[/latex] into the other equation
3. Simplify to get an equation in [latex]x[/latex] and [latex]y[/latex] only

For Trigonometric Equations: When [latex]x(t) = a\cos t[/latex] and [latex]y(t) = b\sin t[/latex]:

1. Solve for [latex]\cos t[/latex] and [latex]\sin t[/latex]: [latex]\frac{x}{a} = \cos t[/latex] and [latex]\frac{y}{b} = \sin t[/latex]
2. Use the Pythagorean identity: [latex]\cos^2 t + \sin^2 t = 1[/latex]
3. Substitute: [latex]\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1[/latex]
4. This produces [latex]\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1[/latex] (an ellipse)

When eliminating the parameter, be careful to preserve any domain restrictions from the original parametric equations.

Graphing Parametric Equations

Key Features to Identify:

• Orientation: The direction along the curve as [latex]t[/latex] increases, shown with arrows
• Starting and ending points: Evaluate at the minimum and maximum values of [latex]t[/latex] in the given interval
• Special points: Where the curve crosses axes or has interesting behavior

Comparing Parametric and Rectangular Graphs: The same curve can be represented both ways. Parametric form shows direction/motion; rectangular form shows the geometric shape. When graphed together, they produce identical curves.

Projectile Motion – A key application of parametric equations:

• Horizontal position: [latex]x = (v_0 \cos\theta)t[/latex]
• Vertical position: [latex]y = -16t^2 + (v_0 \sin\theta)t + h[/latex]

Where [latex]v_0[/latex] is initial velocity, [latex]\theta[/latex] is launch angle, and [latex]h[/latex] is initial height. The [latex]-16[/latex] comes from gravity (in feet per second squared).

Understanding Vectors

A vector is a quantity with both magnitude (size/length) and direction, represented as a directed line segment with an arrow.

• Vector Components:
  • Initial point: Where the vector starts
  • Terminal point: Where the vector ends (arrow tip)
  • Position vector: A vector with initial point at the origin [latex](0, 0)[/latex]
• Vector Notation:
  • Boldface: [latex]\mathbf{v}[/latex] or with arrow: [latex]\vec{v}[/latex]
  • From point [latex]P[/latex] to point [latex]Q[/latex]: [latex]\overrightarrow{PQ}[/latex]
  • Component form: [latex]\langle a, b \rangle[/latex] or [latex]v = a\mathbf{i} + b\mathbf{j}[/latex]
• Position Vector: Given initial point [latex](x_1, y_1)[/latex] and terminal point [latex](x_2, y_2)[/latex], the position vector is: [latex]\mathbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle[/latex]
• Magnitude (Length): For vector [latex]\mathbf{v} = \langle a, b \rangle[/latex]: [latex]|\mathbf{v}| = \sqrt{a^2 + b^2}[/latex]
  • Use the Pythagorean theorem—the magnitude is the distance from initial to terminal point.
• Direction (Angle): The angle [latex]\theta[/latex] from the positive x-axis: [latex]\tan\theta = \frac{b}{a}[/latex], so [latex]\theta = \tan^{-1}\left(\frac{b}{a}\right)[/latex]
  • Check which quadrant the vector is in, as inverse tangent only gives angles in quadrants I and IV.
• Component Form: Every vector can be written as [latex]\mathbf{v} = a\mathbf{i} + b\mathbf{j}[/latex] where [latex]\mathbf{i} = \langle 1, 0 \rangle[/latex] is the horizontal unit vector, [latex]\mathbf{j} = \langle 0, 1 \rangle[/latex] is the vertical unit vector, [latex]a[/latex] is the horizontal component, and [latex]b[/latex] is the vertical component
• Unit Vector: A vector with magnitude 1. To find a unit vector in the direction of [latex]\mathbf{v}[/latex]: [latex]\frac{\mathbf{v}}{|\mathbf{v}|}[/latex]

Writing Vectors in Terms of Magnitude and Direction: [latex]\mathbf{v} = |\mathbf{v}|\cos\theta\mathbf{i} + |\mathbf{v}|\sin\theta\mathbf{j}[/latex]

Where [latex]|\mathbf{v}|[/latex] is the magnitude and [latex]\theta[/latex] is the direction angle. This connects to [latex]x = |\mathbf{v}|\cos\theta[/latex] and [latex]y = |\mathbf{v}|\sin\theta[/latex].

Operations with Vectors

Vector Addition – To find [latex]\mathbf{u} + \mathbf{v}[/latex]:

• Geometrically: Place the initial point of [latex]\mathbf{v}[/latex] at the terminal point of [latex]\mathbf{u}[/latex]. The sum goes from the start of [latex]\mathbf{u}[/latex] to the end of [latex]\mathbf{v}[/latex].
• Algebraically: Add corresponding components [latex]\langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle[/latex]

In [latex]\mathbf{i}, \mathbf{j}[/latex] notation: [latex](a\mathbf{i} + b\mathbf{j}) + (c\mathbf{i} + d\mathbf{j}) = (a+c)\mathbf{i} + (b+d)\mathbf{j}[/latex]

Vector Subtraction – To find [latex]\mathbf{u} - \mathbf{v}[/latex]:

• View as [latex]\mathbf{u} + (-\mathbf{v})[/latex] where [latex]-\mathbf{v}[/latex] reverses the direction of [latex]\mathbf{v}[/latex]
• Algebraically: Subtract corresponding components [latex]\langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle[/latex]

Scalar Multiplication – Multiplying a vector by a constant [latex]k[/latex]:

• Changes the magnitude by factor [latex]|k|[/latex]
• If [latex]k < 0[/latex], reverses direction
• Does not change direction if [latex]k > 0[/latex]

[latex]k\langle a, b \rangle = \langle ka, kb \rangle[/latex]

In [latex]\mathbf{i}, \mathbf{j}[/latex] notation: [latex]k(a\mathbf{i} + b\mathbf{j}) = ka\mathbf{i} + kb\mathbf{j}[/latex]

Properties: Scalar multiplication and vector addition can be combined: [latex]k\mathbf{u} + m\mathbf{v} = k\langle a, b \rangle + m\langle c, d \rangle = \langle ka + mc, kb + md \rangle[/latex]

Dot Product – Multiplying two vectors to get a scalar: [latex]\mathbf{v} \cdot \mathbf{u} = \langle a, b \rangle \cdot \langle c, d \rangle = ac + bd[/latex]

Multiply corresponding components and add the results. The result is a number, not a vector.

Finding the Angle Between Two Vectors using dot product: [latex]\cos\theta = \frac{\mathbf{v}}{|\mathbf{v}|} \cdot \frac{\mathbf{u}}{|\mathbf{u}|}[/latex]

This gives: [latex]\theta = \cos^{-1}\left(\frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{v}||\mathbf{u}|}\right)[/latex]

Key Equations

Position vector	[latex]\mathbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle[/latex]
Vector magnitude	[latex]|\mathbf{v}| = \sqrt{a^2 + b^2}[/latex]
Direction angle	[latex]\tan\theta = \frac{b}{a}[/latex], so [latex]\theta = \tan^{-1}\left(\frac{b}{a}\right)[/latex]
Unit vector	[latex]\frac{\mathbf{v}}{|\mathbf{v}|}[/latex]
Vector in terms of magnitude and direction	[latex]\mathbf{v} = |\mathbf{v}|\cos\theta\mathbf{i} + |\mathbf{v}|\sin\theta\mathbf{j}[/latex]
Components from magnitude and direction	[latex]x = |\mathbf{v}|\cos\theta[/latex] and [latex]y = |\mathbf{v}|\sin\theta[/latex]
Vector addition	[latex]\mathbf{v} + \mathbf{u} = \langle a+c, b+d \rangle[/latex]
Vector subtraction	[latex]\mathbf{v} - \mathbf{u} = \langle a-c, b-d \rangle[/latex]
Scalar multiplication	[latex]k\mathbf{v} = \langle ka, kb \rangle[/latex]
Dot product	[latex]\mathbf{v} \cdot \mathbf{u} = ac + bd[/latex]
Angle between vectors	[latex]\cos\theta = \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{v}||\mathbf{u}|}[/latex]
Projectile motion (horizontal)	[latex]x = (v_0\cos\theta)t[/latex]
Projectile motion (vertical)	[latex]y = -16t^2 + (v_0\sin\theta)t + h[/latex]

Glossary

component form

A way of writing a vector showing its horizontal and vertical parts: [latex]\langle a, b \rangle[/latex] or [latex]a\mathbf{i} + b\mathbf{j}[/latex].

curvilinear path

A curved path or trajectory, as opposed to a straight line; the type of motion described by parametric equations.

dot product

The sum of the products of corresponding components of two vectors; produces a scalar: [latex]\mathbf{v} \cdot \mathbf{u} = ac + bd[/latex].

initial point

The starting point of a vector; the origin [latex](0, 0)[/latex] for a position vector.

magnitude

The length of a vector, calculated using the Pythagorean theorem: [latex]|\mathbf{v}| = \sqrt{a^2 + b^2}[/latex].

orientation

The direction along a parametric curve as the parameter [latex]t[/latex] increases, typically shown with arrows.

parameter

A third variable (often [latex]t[/latex] for time) upon which both [latex]x[/latex] and [latex]y[/latex] depend in parametric equations.

parametric equations

A pair of equations [latex]x = f(t)[/latex] and [latex]y = g(t)[/latex] that define a curve by expressing both coordinates as functions of a parameter.

position vector

A vector with its initial point at the origin [latex](0, 0)[/latex] and terminal point at [latex]\langle a, b \rangle[/latex].

resultant

The vector that results from adding or subtracting two vectors, or from scalar multiplication.

scalar

A quantity with magnitude but no direction; a constant or real number.

scalar multiplication

The operation of multiplying a vector by a scalar, which changes the magnitude and possibly reverses direction: [latex]k\mathbf{v} = \langle ka, kb \rangle[/latex].

standard position

The placement of a vector with initial point at [latex](0, 0)[/latex]; the position vector representation.

terminal point

The ending point of a vector, indicated by the arrowhead.

unit vector

A vector with magnitude 1; [latex]\mathbf{i} = \langle 1, 0 \rangle[/latex] and [latex]\mathbf{j} = \langle 0, 1 \rangle[/latex] are the standard unit vectors.

vector

A quantity with both magnitude and direction, represented as a directed line segment.

vector addition

The operation of combining two vectors by adding corresponding components to produce a resultant vector.