Question

if |a+b|/|a-b|=1, then angle between a and b is???

Answer 1

The angle between $\vec{a}$ and $\vec{b}$ is 90°

Correct question : If $\dfrac{ | \vec{a} + \vec{b}| }{ | \vec{a} - \vec{b}|} = 1$ then angle between $\vec{a}$ and $\vec{b}$ is

Given :

$\dfrac{ | \vec{a} + \vec{b}| }{ | \vec{a} - \vec{b}|} = 1$

To find :

The angle between $\vec{a}$ and $\vec{b}$

Solution :

Step 1 of 2 :

Write down the given equation

Here the given equation is

$\dfrac{ | \vec{a} + \vec{b}| }{ | \vec{a} - \vec{b}|} = 1$

Step 2 of 2 :

Find the angle

$\dfrac{ | \vec{a} + \vec{b}| }{ | \vec{a} - \vec{b}|} = 1$

$\implies | \vec{a} + \vec{b}| = | \vec{a} - \vec{b}|$

${ \implies } {| \vec{a} + \vec{b}|}^{2} = {| \vec{a} - \vec{b}| }^{2}$

${ \implies }(\vec{a} + \vec{b}).(\vec{a} + \vec{b}) = (\vec{a} - \vec{b}).(\vec{a} - \vec{b})$

$\implies (\vec{a}. \vec{a}+ \vec{a}. \vec{b} + \vec{b}. \vec{a} +\vec{b}. \vec{b} )=(\vec{a}. \vec{a} - \vec{a}. \vec{b} - \vec{b}. \vec{a} +\vec{b}. \vec{b} )$

$\implies 2 \vec{a}. \vec{b} + 2\vec{b}. \vec{a} = 0$

$\implies 4 \vec{a}. \vec{b} = 0\: \: \: \bigg[ \: \because \:\vec{a}. \vec{b} = \vec{b}. \vec{a} \bigg]$

$\implies \vec{a}. \vec{b} = 0$

So $\vec{a}$ and $\vec{b}$ are perpendicular to each other

Hence angle between $\vec{a}$ and $\vec{b}$ is 90°

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