Why is the dot product of 2 orthogonal vectors equal to zero? (cosine similarity)

Indent

𝑥·𝑦 = 0

Proof With Pythagorean Theorem

• 𝑎² + 𝑏² = 𝑐²
• ||𝑥̅||² + ||𝑦̅||² = ||𝑥̅ - 𝑦̅||²
• 𝑥̅^T𝑥̅ + 𝑦̅^T𝑦̅ = (𝑥̅ - 𝑦̅)^T(𝑥̅ - 𝑦̅)
• (𝑥₁𝑥₁ + 𝑥₂𝑥₂ + … + 𝑥_𝑛𝑥_𝑛) + (𝑦₁𝑦₁ + 𝑦₂𝑦₂ + … + 𝑦_𝑛𝑦_𝑛) = ([𝑥₁𝑥₁ - 2𝑥₁𝑦₁+ 𝑦₁𝑦₁] + [𝑥₂𝑥₂ - 2𝑥₂𝑦₂+ 𝑦₂𝑦₂] + … + [𝑥_𝑛𝑥_𝑛 - 2𝑥_𝑛𝑦_𝑛+ 𝑦_𝑛𝑦_𝑛])
• 𝑥₁𝑥₁ + 𝑥₂𝑥₂ + … + 𝑥_𝑛𝑥_𝑛 + 𝑦₁𝑦₁ + 𝑦₂𝑦₂ + … + 𝑦_𝑛𝑦_𝑛 = 𝑥₁𝑥₁ - 2𝑥₁𝑦₁+ 𝑦₁𝑦₁ + 𝑥₂𝑥₂ - 2𝑥₂𝑦₂+ 𝑦₂𝑦₂ + … + 𝑥_𝑛𝑥_𝑛 - 2𝑥_𝑛𝑦_𝑛+ 𝑦_𝑛𝑦_𝑛
• 0 = - 2𝑥₁𝑦₁ - 2𝑥₂𝑦₂ - … - 2𝑥_𝑛𝑦_𝑛
• 0 = -2 (𝑥₁𝑦₁ + 𝑥₂𝑦₂ + … + 𝑥_𝑛𝑦_𝑛)
• 0 = 𝑥₁𝑦₁ + 𝑥₂𝑦₂ + … + 𝑥_𝑛𝑦_𝑛
• 0 = 𝑥·𝑦