Question

Given that \[ |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \] which of the following is true?

• A. \( |\vec{a}| = |\vec{b}| \)
• B. \( \vec{a} = \vec{b} \)
• C. \( \vec{a} \perp \vec{b} \)
• D. \( |\vec{a}| = 0 \)

Hint:

If \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \), it implies that the dot product \( \vec{a} \cdot \vec{b} = 0 \), meaning the vectors are perpendicular.

Answer 1

The Correct Option is C

We are given the equation: \[ |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \] Squaring both sides of the equation to eliminate the magnitudes: \[ |\vec{a} + \vec{b}|^2 = |\vec{a} - \vec{b}|^2 \] \[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \] Expanding both sides: \[ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \] Simplifying: \[ 2 \vec{a} \cdot \vec{b} = -2 \vec{a} \cdot \vec{b} \] \[ 4 \vec{a} \cdot \vec{b} = 0 \] \[ \vec{a} \cdot \vec{b} = 0 \] Since the dot product \( \vec{a} \cdot \vec{b} = 0 \), this means that the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular. Thus, the correct answer is: \[ \boxed{\vec{a} \perp \vec{b}} \]