Question

The magnitudes of two vectors are A and B (A>B). If the maximum resultant magnitude of the two vectors is ‘n’ times their minimum resultant magnitude, then \(\frac{A}{B} =\)

• A. \(\frac{n}{n - 1}\)
• B. \(\frac{n + 1}{n}\)
• C. \(\frac{n^2 + 1}{n - 1}\)
• D. \(\frac{n + 1}{n - 1}\)

Hint:

When two vectors are added, the maximum resultant is their sum and the minimum is their difference. Use these identities to build ratios and simplify equations.

Answer 1

The Correct Option is D

Step 1: Use the formulas for maximum and minimum resultant of two vectors.
Maximum resultant: \( R_{\text{max}} = A + B \)
Minimum resultant: \( R_{\text{min}} = A - B \) 
Step 2: According to the question,
\[ \frac{R_{\text{max}}}{R_{\text{min}}} = \frac{A + B}{A - B} = n \] Step 3: Solve the equation.
\[ \frac{A + B}{A - B} = n \Rightarrow A + B = n(A - B) \] \[ A + B = nA - nB \Rightarrow A - nA = -nB - B \Rightarrow A(1 - n) = -B(n + 1) \] \[ \Rightarrow \frac{A}{B} = \frac{n + 1}{n - 1} \] Step 4: Select the correct option.
The value of \(\frac{A}{B}\) is \(\frac{n + 1}{n - 1}\), which matches option (4).