Question

Let three vectors \(\vec a,\vec b\) and \(\vec c\) be such that \(|\vec a|=3,|\vec b|=4,|\vec c|=5\) and each of them is perpendicular to the sum of the other two vectors, then find the value of \(|\vec a+\vec b+\vec c|\).

Hint:

Each vector ⊥ sum of the other two gives a·b+b·c+c·a = 0; expand |a+b+c|².

Answer 1

Step 1: Translating the perpendicularity conditions:
\(\vec a\perp(\vec b+\vec c)\Rightarrow \vec a\cdot\vec b+\vec a\cdot\vec c=0\). Similarly \(\vec b\cdot\vec a+\vec b\cdot\vec c=0\) and \(\vec c\cdot\vec a+\vec c\cdot\vec b=0\).

Step 2: Adding all three equations:
Summing: \(2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)=0\Rightarrow \vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a=0\).

Step 3: Expanding the square of the sum:
\(|\vec a+\vec b+\vec c|^2=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)\).

Step 4: Substituting:
\(=3^2+4^2+5^2+2(0)=9+16+25=50\).

Final Answer:
\[ \boxed{|\vec a+\vec b+\vec c|=\sqrt{50}=5\sqrt2} \]