# Use Schwarz inequality to prove triangle inequality

Dafe

## Homework Statement

Use Schwarz inequality on $$\bar{v} \bullet \bar{w}$$ to prove:

$$||\bar{v} + \bar{w}||^2 \leq (||\bar{v}|| + ||\bar{w}||)^2$$

## Homework Equations

Schwarz inequality:
$$|\bar{v} \bullet \bar{w}| \leq ||\bar{v}|| ||\bar{w}||$$

## The Attempt at a Solution

The way I understand Schwarz inequality is that the product of two unit vectors can not exceed one.
The problem asks me to use that fact to prove that the length of the sum of two vectors does not exceed the sum of the length of two vectors.

I am unable to see a connection, and would appreciate it if someone could push me in the right direction.

Thank you.

$$||\bar{v} + \bar{w}||^2 = \bar{v} \bullet \bar{v} + 2\bar{v} \bullet \bar{w} + \bar{w} \bullet \bar{w} \leq ||\bar{v}||^2 + 2||\bar{v}|| ||\bar{w}|| + ||\bar{w}||^2 = (||\bar{v}|| + ||\bar{w}||)^2$$