These inequalities state the pretty obvious fact that you don't have to travel as far if you go direct.

For the triangle above we may write

Proof: As a vector we may writehenceand

The expression on the right hand side factorises into a perfect square:

Now square root both sides to obtainIt is important to realise thatandcan in general be vectors or numbers. This expression is called the 'forwards' form of the triangle inequality. There is also a 'backwards' form:

The backwards form is quite easy to prove by considering the two casesandseparately.

In either caseis satisfied.