Question

If $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$ for any two vectors, then vectors $\mathbf{a}$ and $\mathbf{b}$ are:

• A. orthogonal vectors
• B. parallel to each other
• C. unit vectors
• D. collinear vectors

Hint:

When the magnitudes of sum and difference of two vectors are equal, the vectors are orthogonal.

Answer 1

The Correct Option is A

Given that $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$, squaring both sides yields $|\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a} - \mathbf{b}|^2$. Expanding this expression, we get $(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})$. This simplifies to $a^2 + 2\mathbf{a} \cdot \mathbf{b} + b^2 = a^2 - 2\mathbf{a} \cdot \mathbf{b} + b^2$. Consequently, $2\mathbf{a} \cdot \mathbf{b} = -2\mathbf{a} \cdot \mathbf{b}$, leading to $\mathbf{a} \cdot \mathbf{b} = 0$. This indicates that the vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal.