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: Geometric View:
Three mutually perpendicular vectors are like the edges of a box with edge lengths 3, 4 and 5.

Step 2: Diagonal:
The sum of the three vectors is the space diagonal of this box, with length $\sqrt{3^2+4^2+5^2}=\sqrt{50}$.

Step 3: Answer:
$\sqrt{50}=5\sqrt2$. Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C) } 5\sqrt{2}} \]