Three vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) are such that \(|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5\) and each of these is perpendicular to the sum of the other two. What is \(|\vec{a}+\vec{b}+\vec{c}|\) ?

Explanation:

Three vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) are such that \(|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5\) and each of these is perpendicular to the sum of the other two.

So, \(\vec{a}\) is perpendicular to \(\vec{b}+ \vec{c}\) so \(\vec{a}.(\vec{b}+ \vec{c})\) = 0 

⇒ \(\vec{a}.\vec{b}+ \vec{a}.\vec{c}\) = 0....(i)

\(\vec{b}\) is perpendicular to \(\vec{a}+ \vec{c}\) so \(\vec{b}.(\vec{a}+ \vec{c})\) = 0 

⇒ \(\vec{b}.\vec{c}+ \vec{b}.\vec{c}\) = 0....(ii)

\(\vec{c}\) is perpendicular to \(\vec{a}+ \vec{b}\) so \(\vec{c}.(\vec{a}+ \vec{b})\) = 0 

⇒ \(\vec{c}.\vec{a}+ \vec{c}.\vec{b}\) = 0....(iii)

Adding (i), (ii), (iii) we get

\(\vec{a}.\vec{b}+ \vec{a}.\vec{c}\) + \(\vec{b}.\vec{c}+ \vec{b}.\vec{c}\) + \(\vec{c}.\vec{a}+ \vec{c}.\vec{b}\) = 0

⇒ \(2(\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a})\) = 0

Now, \(|\vec{a}+\vec{b}+\vec{c}|\)² = \((\vec{a}+\vec{b}+\vec{c}).(\vec{a}+\vec{b}+\vec{c})\)

= \(|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a})\)

= 9 + 16 + 25 + 2(0) = 50

\(|\vec{a}+\vec{b}+\vec{c}|\) = \(5 \sqrt{2}\)

(1) is correct