LessonMathematics · Scalar Product

Scalar Product

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Lesson overview

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Definition and formulas

Geometric definition

u·v = ‖u‖·‖v‖·cos(θ) where θ is the angle between the vectors.

u \cdot v = ||u|| \cdot ||v|| \cdot cos(θ)

Analytic formula

If u = (x, y) and v = (x', y') then u·v = xx' + yy'

u \cdot v = x \cdot x' + y \cdot y'

Perpendicular vectors

u ⊥ v ⟺ u·v = 0.

u ⊥ v \Longleftrightarrow u \cdot v = 0

Properties and applications

Bilinearity and symmetry

u·v = v·u (k·u)·v = k(u·v) (u+v)·w = u·w + v·w

u \cdot v = v \cdot u, (u+v) \cdot w = u \cdot w + v \cdot w

Square of norm

‖u‖² = u·u = x² + y²

||u||^{2} = u \cdot u

Pythagorean theorem

ABC right angle at A ⟺ AB·AC = 0 ⟺ BC² = AB² + AC²

BC^{2} = AB^{2} + AC^{2} (angle droit en A)

Law of cosines

BC² = AB² + AC² − 2·AB·AC·cos(A)

a^{2} = b^{2} + c^{2} - 2bc \cdot cos(A)

Essential references (Morocco Common Core Science)

Problem-solving method

Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.

Common mistakes

Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.

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