Prove Triangle Inequality: ||a|| - ||b|| ≤ ||a - b||

[cscott]

"Use the triangle inequality to prove that:
$$||\vec{a}|| - ||\vec{b}|| \le ||\vec{a} - \vec{b}||$$"

I can start from that expression and prove it true using the Cauchy-Shwarz inequality but I don't think that's what's asked. Any hints?

 

[StatusX]

Do they mean the following triangle inequality:

$$||\vec{a}|| + ||\vec{b}|| \le ||\vec{a} + \vec{b}||$$

because it can be proved from that by picking a and b correctly.

 

[cscott]

Do they mean the following triangle inequality:

$$||\vec{a}|| + ||\vec{b}|| \le ||\vec{a} + \vec{b}||$$

because it can be proved from that by picking a and b correctly.

Yeah but it should be ||a|| + ||b|| >= ||a + b||

 

[river_rat]

Start with
$$||\vec{u} + \vec{v}|| - ||\vec{v}|| \le ||\vec{u}||$$

and choose

$$\vec{u} + \vec{v} = \vec{a}$$

The rest should be pretty self evident after that.

 

[StatusX]

Yeah but it should be ||a|| + ||b|| >= ||a + b||

Right, sorry. So you got the answer?

 

[cscott]

Start with
$$||\vec{u} + \vec{v}|| - ||\vec{v}|| \le ||\vec{u}||$$

and choose

$$\vec{u} + \vec{v} = \vec{a}$$

The rest should be pretty self evident after that.

I got it but how do I justify saying that?