Question

If | a+b | = | a-b | , show that a is perpendicular to b ?

Answer 1

Answer:

square both sides

a^2+b^2+2ab=a^2+b^-2ab

4ab=0

ab=0

if and only if

cos90=0

so a and b perpendicular

Answer 2

Given :

| a+b | = | a-b | .

To Show :

a is perpendicular to b .

Solution :

We know , | a+b | is given by :

$| a+b | =\sqrt{a^2+b^2+2abcos\ \theta}$

Also , | a-b | is given by :

$| a-b | =\sqrt{a^2+b^2-2abcos\ \theta}$

Here , $\theta$ is the angle between a and b .

Equating them , we get :

$\sqrt{a^2+b^2+2abcos\ \theta}=\sqrt{a^2+b^2-2abcos\ \theta}\\\\a^2+b^2+2abcos\ \theta=a^2+b^2-2abcos\ \theta\\\\cos\ \theta=0$

So , $\theta$ is $90^o$ .

Therefore ,  a is perpendicular to b or vise versa .

Learn More :

Vectors

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