Vectors Summary

Trigonometry · Axiom Academy

Let's review how vectors extend trigonometry beyond angles to include magnitude and direction, essential for physics, engineering, and computer graphics. Definition: A mathematical object with both magnitude (length) and direction, represented as v = ⟨a, b⟩ in 2D or v = ⟨a, b, c⟩ in 3D Magnitude: The length of the vector, calculated as ||v|| = √(a² + b²) in 2D or ||v|| = √(a² + b² + c²) in 3D Components: The horizontal and vertical (and depth in 3D) parts that completely describe the vector's position Unit Vector: A vector with magnitude 1, found by dividing by the magnitude: û = v/||v|| Addition: Combine vectors component-wise: u + v = ⟨u₁ + v₁, u₂ + v₂⟩ . Geometrically, place tail of second at head of first Subtraction: Find the vector from v to u: u - v = ⟨u₁ - v₁, u₂ - v₂⟩ Scalar Multiplication: Scale a vector by constant k: k·v = ⟨k·a, k·b⟩ . Changes magnitude but preserves direction (if k > 0) Resultant: The combined effect of multiple vectors, found through repeated addition Definition: Measures alignment between vectors: u·v = u₁v₁ + u₂v₂ + u₃v₃ Geometric Formula: u·v = ||u|| ||v|| cos θ , where θ is the angle between vectors Finding Angles: Rearrange to get θ = cos⁻¹(u·v / (||u|| ||v||)) Orthogonality Test: Vectors are perpendicular when u·v = 0 Projection: The component of u along v is proj v u = ((u·v)/(||v||²))v Definition: Creates a vector perpendicular to both input vectors, calculated using the determinant method

This is the written version of the interactive lesson above. See the full Trigonometry course.