Question

If $|\vec{a} | = 15 , |\vec{b} | = 12$ and $|\vec{a} + \vec{b} | = 20 $ then $|\vec{a} - \vec{b} | = $

• A. $\sqrt{338}$
• B. 338
• C. 769
• D. $\sqrt{769}$

Answer 1

Answer: (A)

$\left|\vec{a}\right|=15, \left|\vec{b}\right|=12, \left|\vec{a} + \vec{b}\right| =20$
$ \left|\vec{a} + \vec{b}\right|^{2} = \left|\vec{a}\right|^{2} +2 \vec{a}.\vec{b} +\left|b\right|^{2}$
$ 2 \vec{a} .\vec{b} =\left(20\right)^{2} -\left(15\right)^{2} -\left(12\right)^{2} $
$= 400 - 225 -144 = 31$
Now, $ \left|\vec{a} - \vec{b}\right|^{2} =\left|\vec{a}\right|^{2} - 2\vec{a} . \vec{b} + \left|\vec{b}\right|^{2} $
$=\left(15\right)^{2} - 31 +\left(12\right)^{2} = 225 - 31 + 144 = 338$
$\therefore \:\:\: \left|\vec{a} - \vec{b}\right| \sqrt{338}$