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