# Use Schwarz inequality to prove triangle inequality

1. Oct 6, 2009

### Dafe

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

3. 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.

2. Oct 6, 2009

### lanedance

start by multiplying out the left hand side then use schcwarz to get to right hand side

3. Oct 6, 2009

### Dafe

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

I think I see the Schwarz in there :)

Thank you lanedance.

4. Nov 24, 2011