At a glance
Pairs · launched with equations of lines, presented at the symposium · one wind, one heading, one runway — hit it
What you are making
Your flight packet holds an airspeed your aircraft can hold, a wind that does not care about your plans, and a destination bearing. Act one is two-dimensional: find the heading that makes your air vector plus the wind vector point along the desired track, and the ground speed that results — arrows doing the algebra. Act two climbs into three dimensions: the final approach is a line in space, , and the runway lies in a plane. You finish with the flight plan: heading and ground speed defended, the approach line written in vector and parametric form, the touchdown point where line meets plane — solved, then checked by substitution — and the descent angle at which you cross it.
Milestones
- The wind triangle drawn before any components appear: air vector, wind vector, ground vector, tip to tail per Adding and Scaling Vectors
- Heading and ground speed computed in components, then checked against the drawing’s rough angles
- The approach line built from your packet’s data in vector and parametric form, per Equations of Lines
- The touchdown point found, then verified by substituting it into the runway plane’s scalar equation
- The descent angle argued with The Dot Product and the plane’s normal
How it is assessed
Per How Marks Work, the reasoning is the product: a landing a few metres off, with every step traceable and the error hunted down, outranks perfect coordinates that appeared from nowhere. The flight plan is presented at The Math Symposium, and the Math Journal entry on what the model ignores — gusts, the wind’s own third dimension, the pilot — completes the evidence.
Success criteria
| Quality | What it looks like in your work |
|---|---|
| Picture first | The wind triangle drawn before components |
| Arrows that add | The heading argued from a vector sum |
| Three forms fluent | The approach line written two ways, correctly |
| A checked landing | The intersection substituted back and verified |
| A stated angle | The descent angle traced to a dot product |
If the line and the plane refuse to meet
Dot the line’s direction vector with the plane’s normal first: if that dot product is zero, your approach runs parallel to the runway — reread the packet, because that is a transcription error, not a mathematics one.
Curriculum connection
C2.3
solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications Sample problem: A plane on a heading of N 27° E has an air speed of 375 km/h. The wind is blowing from the south at 62 km/h. Determine the actual direction of travel of the plane and its ground speed.
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C4.2
recognize that a line in three-space cannot be represented by a scalar equation, and represent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line) Sample problem: Represent the line passing through and with the scalar equations of two intersecting planes, with a vector equation, and with parametric equations.
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C4.7
solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equations) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically Sample problem: Determine the intersection of the perpendicular line drawn from the point to the plane , and determine the distance from point to the plane.
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