Question

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VECTORS. ​

Answer 1

Let the unit vectors be $\hat a\ \ \&\ \ \hat b$ where $a=b=1.$

Let $\theta$ be the angle between these two unit vectors.

Given that,

$\mid\hat a + \hat b\mid = 1\\\\\sqrt{a^2+b^2+2ab\cos\theta}=1$

Since $a=b=1,$

$2(1 + \cos\theta) = 1\\\\\cos\theta=-\dfrac{1}{2}$

Now let's find $\mid\hat a - \hat b\mid$ or $\mid\hat b - \hat a\mid.$

Well, both are same! So,

$\mid\hat a - \hat b\mid\\\\=\sqrt{a^2 + (-b)^2+ 2a(-b)\cos\theta}\\\\=\sqrt{2 + \left(2\cdot-1\cdot-\dfrac{1}{2}\right)}\\\\=\sqrt{2+1}\\\\=\sqrt{3}$

Hence the Proof!