Question

For any two vectors \( \vec{u} \) and \( \vec{v} \), if \( |\vec{u} + \vec{v}| = |\vec{u} - \vec{v}| \) then the angle between them is equal to

• A. \(\frac{\pi}{4}\)
• B. \(\frac{3\pi}{5}\)
• C. \(\frac{\pi}{2}\)
• D. \(\pi\)

Answer 1

The Correct Option is C

If \( |\vec{u} + \vec{v}| = |\vec{u} - \vec{v}| \) then square both sides: \[ |\vec{u} + \vec{v}|^2 = |\vec{u} - \vec{v}|^2 \Rightarrow u^2 + v^2 + 2 \vec{u} \cdot \vec{v} = u^2 + v^2 - 2 \vec{u} \cdot \vec{v} \] \[ \Rightarrow 4 \vec{u} \cdot \vec{v} = 0 \Rightarrow \vec{u} \cdot \vec{v} = 0 \] Thus, angle between vectors = \(90^\circ = \frac{\pi}{2}\)