proving the sum and difference equation for cosine

Question

Suppose that 𝒗 and 𝒘 are unit vectors. If the angle between 𝒗 and 𝒊̂ is 𝛼 and that between 𝒘 and 𝒊̂ is 𝛽, prove that cos(𝛼−𝛽)=cos𝛼cos𝛽+sin𝛼sin𝛽

Answer 1

Immanuel, let's do this one!

v = (cos α)i + (sin α)j

and w = (cos β)i + (sin β)j

cos (α - β) = (v . w) / (||v|| . ||w||)

= ((cos α)i + (sin α)j) . ((cos β)i + (sin β)j) / (1 . 1) . . . . . . . since v and w are unit vectors

= cos α cos β + sin α sin β

Done!