To prove that vectors (a + b)(a - b) = a^2 + b^2 iff

[nishantve1]

Homework Statement

The question says :
Prove that
$(\vec{a} + \vec{b}).(\vec{a} - \vec{b}) = \left | \vec{a} \right |^{2} + \left | \vec{b} \right |^{2}$
if and only if $\vec{a} \perp \vec{b}$

Homework Equations

This is known that
$\left | \vec{a} + \vec{b} \right | = \left | \vec{a} - \vec{b} \right |$
if $\vec{a} \perp \vec{b}$

The Attempt at a Solution

I tried substituting that and then using Cauchy–Schwarz inequality but somehow I can't open up the absolute brackets .
Thanks in advance

 

[Saitama]

You can simply use the distributive property.

EDIT: Are you sure that you have copied down the question correctly? I guess the RHS should be $$\left | \vec{a} \right |^{2} - \left | \vec{b} \right |^{2}$$

 

[nishantve1]

@Pranav The question is correct

 

[Saitama]

@Pranav The question is correct

Substitute $\vec{a}=\hat{i}$ and $\vec{b}=\hat{j}$. The relation doesn't satisfy this.

 

[jfgobin]

That doesn't sound right: take the following example:

$\vec{a}=\left ( 1 , 0\right )$
$\vec{b}=\left ( 0 , -0.1\right )$

$\left ( \vec{a} + \vec{b} \right ) \cdot \left ( \vec{a} - \vec{b} \right ) = 0.99$

but

$\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 = 1.01$

J.

 

[HallsofIvy]

As several people have told you now, the correct relation is
$$(\vec{a}+ \vec{b})\cdot(\vec{a}- \vec{b})= |\vec{a}|^2- |\vec{b}|^2$$
not what you wrote. You should have realized that from the usual numerical "difference of squares" formula.

As to how to prove it, what "tools" do have to? You write as if you had never seen vectors before but that can't be true if you are expected to do a probolem like this!
How are you working with vectors? In terms of components? Are you allowed to assume two or three dimensions? Or in terms of "length and direction"?

How has the dot product been defined? As "$<a_1, a_2, \cdot\cdot\cdot, a_n>\cdot<b_1, b_2, \cdot\cdot\cdot, b_n>= a_1b_1+ a_2b_2+ \cdot\cdot\cdot+ a_nb_n$? Or as $|\vec{a}||\vec{b}|sin(\theta)$ where $\theta$ is the angle between $\vec{a}|$ and $\vec{b}$?

Either can be used but the proof depends on which you are using.

 

[D H]

Prove that
$(\vec{a} + \vec{b}).(\vec{a} - \vec{b}) = \left | \vec{a} \right |^{2} + \left | \vec{b} \right |^{2}$
if and only if $\vec{a} \perp \vec{b}$

This is obviously false. Double check the question. It is probably asking you to prove that $(\vec a + \vec b) \cdot (\vec a + \vec b) = ||\vec a||^2 + ||\vec b||^2$ iff $\vec{a} \perp \vec{b}$.

 

[mfb]

$$(\vec{a}+ \vec{b})\cdot(\vec{a}- \vec{b})= |\vec{a}|^2- |\vec{b}|^2$$

But this is true for all a,b.

My guess: Show that
$$(\vec{a}+ \vec{b})\cdot(\vec{a}+ \vec{b})= |\vec{a}|^2+ |\vec{b}|^2$$
is true if and only if $\vec{a} \perp \vec{b}$.

Edit: Same minute :(