Prove that the cosine formula for triangles is equivalent to the definition of the scalar product.

Explanation

Let two sides of a triangle represent the two vectors `vec(a) and vec(b)` .
Then the third side is `vec(c ) = vec(a) - vec(b)`
`(vec(a) - vec(b)).(vec(a) - vec(b)) = vec(c ) . vec(c ) `
`a^(2) - vec(a). vec(b)- vec(b)-vec(a)+b^(2) = c^(2)`
`a^(2) +b^(2) - 2 vec(a) . vec(b) = c^(2)`
`vec(a).vec(b) = ab cos theta `
where ` theta ` is the angle between the two vectors `vec(a) and vec(b)` .
This is the cosine law .