Unit 9: Vectors and Complex Numbers

Precalculus Honors | Standards PC.NR.2 & PC.NR.3

Brian Risden

Unit Overview

We will explore the geometry and algebra of vectors and complex numbers, focusing on their representations and applications.

• Vectors (PC.NR.2) — Representing directed line segments, performing operations (component-wise and graphically), and modeling force and velocity.
• Complex Numbers (PC.NR.3) — Representing numbers on the complex plane, converting between rectangular and polar forms, and interpreting operations geometrically.

Vectors: Representation & Modeling

Vector Basics

A vector represents a quantity with both magnitude (length) and direction.

Form	Notation	Formula
Component Form	\(\mathbf{v} = \langle x, y \rangle\)	\(x = v_2 - v_1, y = v_2 - v_1\)
Magnitude	\(\|\mathbf{v}\|\)	\(\sqrt{x^2 + y^2}\)
Direction Angle	\(\theta\)	\(\tan \theta = \frac{y}{x}\)

Interactive: Vector Component Explorer

viewof v_x = Inputs.range([-10, 10], {value: 4, step: 1, label: "x-component"})
viewof v_y = Inputs.range([-10, 10], {value: 3, step: 1, label: "y-component"})

v_mag = Math.sqrt(v_x**2 + v_y**2)
v_ang = Math.atan2(v_y, v_x) * 180 / Math.PI

Plot.plot({
  aspectRatio: 1,
  x: {domain: [-11, 11], grid: true},
  y: {domain: [-11, 11], grid: true},
  marks: [
    Plot.arrow([[0, 0], [v_x, v_y]], {stroke: "#332288", strokeWidth: 4}),
    Plot.text([[v_x, v_y + 1]], {text: [`v = ⟨${v_x}, ${v_y}⟩`], fontSize: 20, fill: "#332288"}),
    Plot.text([[-9, 9]], {
      text: [`Magnitude: ${v_mag.toFixed(2)} | Angle: ${v_ang.toFixed(1)}°`], 
      fontSize: 22, fontWeight: "bold", fill: "#117733"
    })
  ]
})

Vector Addition & Resultants

To find the resultant vector \(\mathbf{u} + \mathbf{v}\):

1. Algebraically: Add corresponding components: \(\langle u_1+v_1, u_2+v_2 \rangle\).
2. Geometrically: Use the Tail-to-Head method or the Parallelogram Law.

Scalar Multiplication: Multiplying a vector by \(k\) changes its magnitude (and direction if \(k < 0\)) but stays on the same line.

Interactive: Addition & Parallelogram Law

viewof u_x = Inputs.range([-5, 5], {value: 3, step: 1, label: "u_x"})
viewof u_y = Inputs.range([-5, 5], {value: 1, step: 1, label: "u_y"})
viewof w_x = Inputs.range([-5, 5], {value: 1, step: 1, label: "v_x"})
viewof w_y = Inputs.range([-5, 5], {value: 4, step: 1, label: "v_y"})

r_x = u_x + w_x
r_y = u_y + w_y

Plot.plot({
  aspectRatio: 1,
  x: {domain: [-1, 10], grid: true},
  y: {domain: [-1, 10], grid: true},
  marks: [
    Plot.arrow([[0, 0], [u_x, u_y]], {stroke: "#117733", strokeWidth: 3}), // u
    Plot.arrow([[0, 0], [w_x, w_y]], {stroke: "#44AA99", strokeWidth: 3}), // v
    // Parallelogram dashed lines
    Plot.line([[u_x, u_y], [r_x, r_y]], {stroke: "#aaa", strokeDasharray: "4"}),
    Plot.line([[w_x, w_y], [r_x, r_y]], {stroke: "#aaa", strokeDasharray: "4"}),
    // Resultant
    Plot.arrow([[0, 0], [r_x, r_y]], {stroke: "#AA4499", strokeWidth: 5}),
    Plot.text([[r_x, r_y + 0.5]], {text: ["Resultant u+v"], fill: "#AA4499", fontSize: 20})
  ]
})

Practice: PC.NR.2

1. Find the component form of a vector with magnitude 10 and direction angle \(150^\circ\).

\(x = 10 \cos(150^\circ) = -5\sqrt{3}, \quad y = 10 \sin(150^\circ) = 5\) \(\rightarrow\) \(\langle -5\sqrt{3}, 5 \rangle\)

2. Given \(\mathbf{u} = \langle 2, -3 \rangle\) and \(\mathbf{v} = \langle 5, 4 \rangle\), find \(2\mathbf{u} - \mathbf{v}\).

\(\langle 4, -6 \rangle - \langle 5, 4 \rangle = \langle 4-5, -6-4 \rangle\) \(\rightarrow\) \(\langle -1, -10 \rangle\)

3. An airplane flies at 400 mph at \(N 30^\circ E\). A wind blows at 50 mph from the West. Find the resultant velocity.

\(\mathbf{p} = \langle 400\sin 30, 400\cos 30 \rangle = \langle 200, 346.4 \rangle\) \(\mathbf{w} = \langle 50, 0 \rangle \rightarrow \mathbf{r} = \langle 250, 346.4 \rangle\) \(\text{Mag} \approx 427.2 \text{ mph}\)

Complex Numbers in the Complex Plane

Rectangular vs. Polar Form

The complex number \(z = a + bi\) can be mapped to \((a, b)\) on the complex plane.

Form	Representation	Conversion
Rectangular	\(z = a + bi\)	\(a = r \cos \theta, b = r \sin \theta\)
Polar	\(z = r(\cos \theta + i \sin \theta)\)	\(r = \sqrt{a^2 + b^2}, \tan \theta = \frac{b}{a}\)

Insight: These represent the same number because they describe the same point \((a, b)\) using either Cartesian coordinates or distance/angle from the origin.

Interactive: Complex Form Converter

viewof c_a = Inputs.range([-10, 10], {value: 3, step: 1, label: "Real (a)"})
viewof c_b = Inputs.range([-10, 10], {value: 3, step: 1, label: "Imaginary (bi)"})

c_r = Math.sqrt(c_a**2 + c_b**2)
c_theta = Math.atan2(c_b, c_a) * 180 / Math.PI

html`<div style="font-size: 26px; padding: 20px; background: #fdfdfd; border-radius: 10px;">
  <p><span style="color:#332288"><b>Rectangular:</b></span> ${c_a} + ${c_b}i</p>
  <p><span style="color:#117733"><b>Polar:</b></span> ${c_r.toFixed(2)} [cos(${c_theta.toFixed(1)}°) + i sin(${c_theta.toFixed(1)}°)]</p>
</div>`

Plot.plot({
  aspectRatio: 1,
  x: {domain: [-11, 11], grid: true, label: "Real"},
  y: {domain: [-11, 11], grid: true, label: "Imaginary"},
  marks: [
    Plot.dot([[c_a, c_b]], {r: 6, fill: "#AA4499"}),
    Plot.line([[0, 0], [c_a, c_b]], {stroke: "#aaa", strokeDasharray: "3"}),
    Plot.text([[c_a, c_b + 0.8]], {text: [`${c_a} + ${c_b}i`], fill: "#AA4499", fontSize: 20})
  ]
})

Operations in Polar Form

Polar form makes multiplication and division much simpler:

Multiplication: Multiply \(r\)’s, Add \(\theta\)’s \(z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]\)

Division: Divide \(r\)’s, Subtract \(\theta\)’s \(\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]\)

Geometric Interpretations (PC.NR.3.2)

• Addition: Follows the parallelogram rule (identical to vectors).
• Conjugation: A reflection over the Real (\(x\)) axis.
• Multiplication by \(i\): A \(90^\circ\) counter-clockwise rotation.
• Multiplication by \(r(\cos \theta + i \sin \theta)\): A scaling by \(r\) and rotation by \(\theta\).

Practice: PC.NR.3

1. Write \(z = -2 + 2i\sqrt{3}\) in polar form.

\(r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = 4\) \(\theta = \text{atan2}(2\sqrt{3}, -2) = 120^\circ\) \(z = 4(\cos 120^\circ + i \sin 120^\circ)\)

2. Find \(z_1 z_2\) for \(z_1 = 3(\cos 40^\circ + i \sin 40^\circ)\) and \(z_2 = 2(\cos 20^\circ + i \sin 20^\circ)\).

\(r = 3 \cdot 2 = 6, \quad \theta = 40^\circ + 20^\circ = 60^\circ\) \(6(\cos 60^\circ + i \sin 60^\circ) = 3 + 3i\sqrt{3}\)

3. Geometrically, what happens to \(z\) when you find its conjugate \(\bar{z}\)?

The imaginary part changes sign: \((a, b) \rightarrow (a, -b)\). This results in a reflection across the horizontal (real) axis.

Summary

Operation	Algebra	Geometry
Vector Addition	\(\langle x_1+x_2, y_1+y_2 \rangle\)	Parallelogram/Head-to-Tail
Complex Addition	\((a+c) + (b+d)i\)	Parallelogram
Complex Mult.	\(r_1r_2, \theta_1+\theta_2\)	Rotation + Scaling
Conjugation	\(a - bi\)	Reflection over Real Axis

\(z = r(\cos \theta + i \sin \theta) \quad \mathbf{v} = \langle \|\mathbf{v}\|\cos \theta, \|\mathbf{v}\|\sin \theta \rangle\) ```