From Two Points to a Unit Vector
A lightboard-style walkthrough shows how to build a vector from two points in 3D space, compute its magnitude, and normalize it into a unit vector. Starting from P(1,-2,4) and Q(3,1,-2), each algebraic step is written out clearly, with color coding to separate routine work, key annotations, and the final answer. Useful for multivariable calculus students learning vector construction and normalization for the first time.
Narrated · 16:9 · Preview before teaching · automatic layout checks do not establish subject accuracy
The prompt that made it
Create a 40-second educational mathematics video in a clean whiteboard/lightboard style. SOURCE: Module 1, Lesson 1.1, Worked Example 1 Title: From Two Points to a Unit Vector TARGET: University-level Multivariable Calculus students. OBJECTIVE: Demonstrate the complete process of going from two points in 3D space to: 1. The vector between the points 2. Its magnitude 3. Its unit vector VISUAL STYLE: - Professional university mathematics lesson. - Clean whiteboard/lightboard aesthetic. - Minimal visual clutter. - Mathematics is the primary visual. - Animate equations as if being written on a board. - Use smooth mathematical transformations. - No characters, avatars, stock imagery, or decorative graphics. - Keep calculations visible as the solution develops. COLOR RULES: - BLACK = standard mathematical work - BLUE = annotations, labels, emphasis and checks - RED = final answer These colors must be applied consistently. MATHEMATICAL CONTENT: P = (1, −2, 4) Q = (3, 1, −2) SCENE 1 — INTRO | 0:00–0:06 Display the title: From Two Points to a Unit Vector Then show: P = (1, −2, 4) Q = (3, 1, −2) Narration: "Given two points in space, let's find the vector between them, its length, and its unit vector." SCENE 2 — VECTOR | 0:06–0:20 Write in BLACK: →PQ = Q − P Then show: →PQ = ⟨3−1, 1−(−2), −2−4⟩ Then simplify: →PQ = ⟨2, 3, −6⟩ Use BLUE to underline or briefly emphasize: 1 − (−2) = 3 Narration: "Subtract the tail from the head. This gives the vector two, three, negative six." SCENE 3 — MAGNITUDE | 0:20–0:29 Write in BLACK: ‖PQ‖ = √(2² + 3² + (−6)²) Then transform: = √49 = 7 Narration: "Now find its magnitude. Square the components, add them, and take the square root. The length is seven." Keep the vector visible. SCENE 4 — UNIT VECTOR | 0:29–0:38 Write: u = →PQ / ‖→PQ‖ Then: u = ⟨2, 3, −6⟩ / 7 Then reveal the final result: u = ⟨2/7, 3/7, −6/7⟩ The final answer must be RED. Draw a RED box around the final unit vector. Narration: "Finally, divide every component by the magnitude. The unit vector is two-sevenths, three-sevenths, negative six-sevenths." SCENE 5 — END | 0:38–0:40 Keep only the final result prominent: u = ⟨2/7, 3/7, −6/7⟩ Add a small BLUE check mark. IMPORTANT MATHEMATICAL ACCURACY: - Preserve every number exactly. - Preserve every negative sign. - Preserve parentheses around negative values. - Use proper vector notation. - Use proper fractions, not decimal approximations. - Do not invent additional mathematical steps. - Do not change the example. - Do not replace equations with ordinary text. - Ensure all equations are mathematically correct and clearly readable. AUDIO: Use the provided narration exactly as written above. Use a clear, calm university-level instructional voice. Keep narration synchronized with the appearance of each mathematical step. FINAL LOOK: The result should feel like a concise 40-second excerpt from a professional university calculus course, with the mathematical working appearing progressively on a whiteboard.
Make the next one in this series
- Show how to find a vector's direction angles from its components
- Demonstrate finding a point a given distance along a vector's direction
- Extend the example to compute the dot product and angle between two vectors
