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

We are given that $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$. Squaring both sides: \[ |\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a} - \mathbf{b}|^2 \] Expanding both sides: \[ (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \] This simplifies to: \[ a^2 + 2\mathbf{a} \cdot \mathbf{b} + b^2 = a^2 - 2\mathbf{a} \cdot \mathbf{b} + b^2 \] Thus, $2\mathbf{a} \cdot \mathbf{b} = -2\mathbf{a} \cdot \mathbf{b}$, which gives $\mathbf{a} \cdot \mathbf{b} = 0$. Therefore, the vectors are orthogonal.