gen-define-vectors-and-dot-products-and-give-one-short-worked-ex-6f83

Define vectors and dot products and give one short worked example.

# Vectors and Dot Products ## Vector A **vector** is a mathematical object with both magnitude (length) and direction. In coordinate form, a vector is written as an ordered list of numbers. **Example notation:** **v** = ⟨3, 4⟩ or **v** = 3**i** + 4**j** ## Dot Product The **dot product** (or scalar product) of two vectors is an operation that produces a scalar (single number). For vectors **a** = ⟨a₁, a₂⟩ and **b** = ⟨b₁, b₂⟩: **Formula:** **a** · **b** = a₁b₁ + a₂b₂ *(For 3D: **a** · **b** = a₁b₁ + a₂b₂ + a₃b₃)* --- ## Worked Example Find the dot product of **u** = ⟨2, -3⟩ and **v** = ⟨4, 1⟩ **Solution:** **u** · **v** = (2)(4) + (-3)(1) = 8 - 3 = **5** The dot product equals 5 (a scalar).