Question

If two vectors \( \mathbf{a} \) and \( \mathbf{b} \) satisfy the equation:

\[ \frac{|\mathbf{a} + \mathbf{b}| + |\mathbf{a} - \mathbf{b}|}{|\mathbf{a} + \mathbf{b}| - |\mathbf{a} - \mathbf{b}|} = \sqrt{2} + 1, \]

then the value of

\[ \frac{|\mathbf{a} + \mathbf{b}|}{|\mathbf{a} - \mathbf{b}|} \]

is equal to:

• A. \( 1 + \sqrt{2} \)
• B. \( 2 + 4\sqrt{2} \)
• C. \( 1 + 2\sqrt{2} \)
• D. \( 3 + 2\sqrt{2} \)

Hint:

When solving problems involving vector magnitudes, break down the equations into manageable parts by substituting for magnitudes and using basic algebraic techniques.

Answer 1

The Correct Option is A

The equation is given as: \[ \frac{|\mathbf{a} + \mathbf{b}| + |\mathbf{a} - \mathbf{b}|}{|\mathbf{a} + \mathbf{b}| - |\mathbf{a} - \mathbf{b}|} = \sqrt{2} + 1. \] Let \( x = |\mathbf{a} + \mathbf{b}| \) and \( y = |\mathbf{a} - \mathbf{b}| \). The equation simplifies to: \[ \frac{x + y}{x - y} = \sqrt{2} + 1. \] Cross-multiplying yields: \[ (x + y) = (\sqrt{2} + 1)(x - y). \] Expanding the right side: \[ x + y = (\sqrt{2} + 1)x - (\sqrt{2} + 1)y. \] Collecting like terms: \[ x + y + (\sqrt{2} + 1)y = (\sqrt{2} + 1)x. \] This simplifies to: \[ x + y(1 + \sqrt{2}) = (\sqrt{2} + 1)x. \] Solving for \( \frac{x}{y} \) (which represents \( |\mathbf{a} + \mathbf{b}| / |\mathbf{a} - \mathbf{b}| \)) gives: \[ \frac{x}{y} = 1 + \sqrt{2}. \] Therefore, the correct answer is \( 1 + \sqrt{2} \), corresponding to option (1).