Question

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of \( |\vec{a} + \vec{b} + \vec{c}|^2 \) is:

• A. \( |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 \)
• B. \( |\vec{a}| + |\vec{b}| + |\vec{c}| \)
• C. \( 2(|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2) \)
• D. \( \frac{1}{2}(|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2) \)
• E. \( 0 \)

Hint:

This geometric condition essentially means that the vectors \( \vec{a}, \vec{b}, \vec{c} \) behave like the axes of a coordinate system (mutually orthogonal) in terms of their dot products summing to zero.

Answer 1

The Correct Option is A

Concept: The condition "each is perpendicular to the sum of the other two" gives: \( \vec{a} \cdot (\vec{b} + \vec{c}) = 0 \), \( \vec{b} \cdot (\vec{a} + \vec{c}) = 0 \), and \( \vec{c} \cdot (\vec{a} + \vec{b}) = 0 \).

Step 1: Expand the dot product conditions.
1. \( \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \) 2. \( \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0 \) 3. \( \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \)

Step 2: Sum the three equations.
Adding all three equations together: \[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \] This implies the sum of the cyclic dot products is zero.

Step 3: Evaluate the magnitude square.
The standard identity for the square of a vector sum is: \[ |\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \] Substituting the result from
Step 2: \[ |\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 0 \]