Question

If \(|\overset{⃗}{a}| = 3\), \(|\overset{⃗}{b}| = 4\), \(|\overset{⃗}{c}| = 5\) such that each vector is perpendicular to the sum of the other two, then \(|\overset{⃗}{a}+\overset{⃗}{b}+\overset{⃗}{c}|\) is equal to

• A. \(5\sqrt{3}\)
• B. \(10\sqrt{2}\)
• C. \(5\sqrt{2}\)
• D. \(4\sqrt{3}\)

Hint:

The conditions force every pair of vectors to be perpendicular.

Answer 1

The Correct Option is C

Step 1: Translate the Conditions:
\(\vec a\cdot(\vec b+\vec c)=0\), \(\vec b\cdot(\vec a+\vec c)=0\), \(\vec c\cdot(\vec a+\vec b)=0\). Write \(x=\vec a\cdot\vec b,\ y=\vec b\cdot\vec c,\ z=\vec c\cdot\vec a\). Then \(x+z=0,\ x+y=0,\ y+z=0\).

Step 2: Solve:
Adding all three gives \(x+y+z=0\), so \(x=y=z=0\). All pairs are perpendicular.

Step 3: Magnitude:
\[ |\vec a+\vec b+\vec c|^2=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(x+y+z)=9+16+25+0=50 \]

Step 4: Result:
\(|\vec a+\vec b+\vec c|=\sqrt{50}=5\sqrt2\). Option (A) \(5\sqrt3\) would need a squared length of 75, and (B) \(10\sqrt2\) of 200.

Final Answer:
The magnitude is \(5\sqrt2\), option (C). \[ \boxed{\text{(C) } 5\sqrt{2}} \]