ベクトルは必ず得点源にすべし【共通テスト】

Main ideas / lessons

• Vectors can be used as a “scoring power source” in the Japanese Common Test by breaking geometric problems into vector equations (including those from roughly the past 20 years).
• Core strategy:
  1. Choose appropriate reference vectors (aligned coordinate viewpoints).
  2. Express unknown vectors (e.g., target points) as linear combinations of those reference vectors.
  3. Determine coefficients using geometric conditions translated into equations (e.g., internal/external division, parallelism, perpendicularity, coplanarity, etc.).
  4. Use inner products to connect vectors with angles (e.g., (\cos\theta), (\sin\theta)) and other geometric quantities (e.g., triangle area, regular polygon lengths).

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Methodology / instruction-style steps (explicit)

1) Choose “reference vectors” (aligned viewpoints)

• 2D (plane) problems:
  • Pick two reference vectors that act like aligned axes/standards (e.g., (\overrightarrow{OA}) and (\overrightarrow{OB})).
  • Ensure the “viewpoint alignment” is consistent:
    • If a condition uses a point that becomes the viewpoint, use that point as the reference origin appropriately.
• 3D (space) problems:
  • Pick three reference vectors (e.g., from a common origin (O)).

Goal: With properly aligned reference vectors, any vector in the problem can be expressed as a linear combination of the reference vectors.

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2) Translate geometric statements into vector equations

The narrator describes “five patterns” as the main toolkit.

A. Point on a line / segment (internal division)

• If (P) lies on line (AB), define a vector expression for (\overrightarrow{OP}) in terms of (\overrightarrow{OA}) and (\overrightarrow{OB}) such that:
  • The coefficients correspond to the division ratio on (AB).
• Typical idea:
  • When (P) divides (AB) with an internal ratio, the coefficients in the linear combination can be arranged so they sum to 1.

B. External division

• If (P) divides (AB) externally:
  • Use the same linear-combination approach, but interpret it as internal division with one coefficient negative.
• Effect:
  • The coefficient sum follows a rule, but may produce values like 1 or sign changes depending on the ratio convention.

C. Perpendicularity / orthogonality via inner product

• Perpendicularity is encoded using:
  • (\vec{u}\cdot \vec{v} = 0)

D. Point on a plane (coplanarity)

• If (P) lies on plane (ABC):
  • Encode coplanarity by requiring the vector coefficients (when decomposed using (\overrightarrow{OA},\overrightarrow{OB},\overrightarrow{OC}) relative to the viewpoint alignment) to satisfy the plane condition.
• Narrator’s version:
  • Express (\overrightarrow{OP}) as a combination of plane basis vectors so that it matches the “plane representation” with coefficients summing appropriately.

E. Line perpendicular to a plane

• If line (PQ) is perpendicular to plane (ABC), then:
  • The vector along the line (e.g., (\overrightarrow{PQ})) is perpendicular to two independent direction vectors in the plane.
• Concrete approach:
  • Choose two plane direction vectors such as (AB) and (AC).
  • Impose:
    • (\overrightarrow{PQ}\cdot \overrightarrow{AB} = 0)
    • (\overrightarrow{PQ}\cdot \overrightarrow{AC} = 0)
  • Solve the resulting equations for unknowns.

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3) Use the inner product for angle and length relations

• Main engine:

  • [ \overrightarrow{OA}\cdot \overrightarrow{OB} = |\overrightarrow{OA}|\,|\overrightarrow{OB}|\cos\theta ]

• Emphasized meaning:

  • The dot product corresponds to a projection length relationship.
• Key consequences:
  • If vectors are perpendicular, then:
    • (\overrightarrow{OA}\cdot \overrightarrow{OB} = 0)
  • Otherwise:
    • Use the dot product to compute (\cos\theta), then derive other trig values as needed.

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4) Inner product to compute triangle area

• The narrator recalls an area formula for triangle (OAB) using vectors.
• The area depends on:
  • (|\overrightarrow{OA}|\,|\overrightarrow{OB}|),
  • (\sin\theta),
  • and therefore on the dot product via (\sin\theta) obtained from (\cos\theta).
• Caution:
  • The expression involves squaring terms, so dimensional consistency matters.

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5) Solve Common Test vectors by coefficient matching

• After expressing unknown vectors in the reference-vector basis:
  • Compare coefficients (or solve a system) to determine unknown scalars.
• If additional geometric constraints exist (perpendicular/parallel/coplanar):
  • Convert them into dot-product or coefficient-sum equations, then solve.

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Example concepts shown (what they illustrate)

• Midpoint/internal division example (triangle (ABC)):
  • Construct (F) as the intersection of lines defined by division rules (e.g., (BD) and (CE)).
  • Express (\overrightarrow{AF}) using (\overrightarrow{AB}) and (\overrightarrow{AC}) by:
    • Choosing reference vectors at a viewpoint (aligned at (A)),
    • Expressing intermediate vectors (e.g., (\overrightarrow{AD}), (\overrightarrow{AE})) in terms of (\overrightarrow{AB}) and (\overrightarrow{AC}),
    • Using internal division coefficient constraints (sums like 1) to solve coefficients.
• 3D regular polyhedron example (regular dodecahedron context):
  • Select reference vectors from a shared origin and use parallel/perpendicular relations to compute required inner products.
• Circle + inner product example (Common Test problem):
  • Use that on a circle with given radius and radius-center at (O), dot products relate to angles between radius vectors.
  • Solve parameters via:
    • (\cos\theta) from the dot product,
    • perpendicularity constraints using dot product (=0).
• Range determination via geometric region:
  • When vector algebra yields a sign/region condition:
    • Interpret it geometrically to determine which side of a line/plane the point lies on based on a derived scalar’s sign (e.g., “positive/negative” corresponds to region D1 vs D2).

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Sources / speakers

• Hajime Kono (Starry) — narrator/presenter of the video.