Question

Let ${\displaystyle \overrightarrow{a},\overrightarrow{b},\overrightarrow{c}}$ be three vectors of magnitudes 3, 4 and 5 respectively. If each one is petpendicular to the sum of the other two vectors, then ${\displaystyle |\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|}$ =

Options

• 5

• ${\displaystyle 3\sqrt{2}}$

• ${\displaystyle 5\sqrt{2}}$

• 12

Answer 1

${\displaystyle 5\sqrt{2}}$

Explanation:

We have ${\displaystyle \left|\overrightarrow{a}\right|}$ = 3, ${\displaystyle \left|\overrightarrow{b}\right|}$ = 4 and ${\displaystyle \left|\overrightarrow{c}\right|}$ = 5.

It is given that ${\displaystyle \overrightarrow{a}\perp (\overrightarrow{b}+\overrightarrow{c}),\overrightarrow{b}\perp (\overrightarrow{c}+\overrightarrow{a})}$ and ${\displaystyle \overrightarrow{c}\perp (\overrightarrow{a}+\overrightarrow{b})}$

⇒ ${\displaystyle \overrightarrow{a}.(\overrightarrow{b}+\overrightarrow{c})}$ = 0, ${\displaystyle \overrightarrow{b}.(\overrightarrow{c}+\overrightarrow{a})}$ = 0, ${\displaystyle \overrightarrow{c}.(\overrightarrow{a}+\overrightarrow{b})}$ = 0

⇒ ${\displaystyle \overrightarrow{a}.\overrightarrow{b}+\overrightarrow{a} .\overrightarrow{c}}$ = 0, ${\displaystyle \overrightarrow{b}.\overrightarrow{c}+\overrightarrow{b}.\overrightarrow{a}}$ = 0, ${\displaystyle \overrightarrow{c}.\overrightarrow{a}+\overrightarrow{c}.\overrightarrow{b}}$ = 0

Adding all these, we get, ${\displaystyle 2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a})}$ = 0

⇒ ${\displaystyle \overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}}$ = 0

Now, ${\displaystyle {|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+{\left|\overrightarrow{c}\right|}^2+2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a})}$ = 3² + 4² + 5² + 0 = 50

⇒ ${\displaystyle |\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|=\sqrt{50}=5\sqrt{2}}$.