A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Geometry and algebra
CSCA Mathematics · Lesson 12 of 16Vectors
Vecteurs · المتجهات
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A vector has a size and a direction. In coordinates, every vector operation becomes simple arithmetic.
Key ideas
- a = (x₁, y₁), b = (x₂, y₂): a + b = (x₁ + x₂, y₁ + y₂) and ka = (kx₁, ky₁).
- Length: |a| = √(x₁² + y₁²).
- Dot product: a·b = x₁x₂ + y₁y₂ = |a||b| cos θ.
- a ⊥ b ⇔ a·b = 0; a ∥ b ⇔ x₁y₂ − x₂y₁ = 0.
Formulas to know
- cos θ = a·b / (|a||b|)
- AB = (x_B − x_A, y_B − y_A)
- |a + b|² = |a|² + 2a·b + |b|²
Worked example
a = (3, 4) and b = (4, −3). Find |a| and a·b. Are they perpendicular?
- Length: |a| = √(9 + 16) = 5
- Dot: a·b = 3·4 + 4·(−3) = 0
- Conclude: a·b = 0, so a ⊥ b
Answer: |a| = 5, a·b = 0: perpendicular
Practice questions
1. a = (1, 2), b = (3, −1). a + 2b = ?
(7, 0) ✓ · (4, 1) · (5, 3) · (7, 4)
(1 + 6, 2 − 2) = (7, 0).
2. For which k are (2, k) and (1, 3) parallel?
k = 3 · k = 6 ✓ · k = −6 · k = 2/3
2·3 − k·1 = 0, so k = 6.
3. Angle between (1, 0) and (1, 1)?
30° · 45° ✓ · 60° · 90°
cos θ = 1/√2, θ = 45°.