Question

If $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$, then $\mathbf{a}$ and $\mathbf{b}$ are

• A. parallel
• B. perpendicular
• C. at angle $45^\circ$
• D. at angle $60^\circ$

Hint:

$|\mathbf{a}+\mathbf{b}| = |\mathbf{a}-\mathbf{b}|$ implies $\mathbf{a} \perp \mathbf{b}$. Geometrically, this is the condition for diagonals of a rectangle to be equal.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Square both sides and expand using the dot product.
Step 2: Detailed Explanation:
$|\mathbf{a}+\mathbf{b}|^2 = |\mathbf{a}-\mathbf{b}|^2$
$\Rightarrow |\mathbf{a}|^2+2\mathbf{a}\cdot\mathbf{b}+|\mathbf{b}|^2 = |\mathbf{a}|^2-2\mathbf{a}\cdot\mathbf{b}+|\mathbf{b}|^2$
$\Rightarrow 4\mathbf{a}\cdot\mathbf{b} = 0 \Rightarrow \mathbf{a}\cdot\mathbf{b} = 0$.
Step 3: Final Answer:
$\mathbf{a}$ and $\mathbf{b}$ are perpendicular.