Question

The ratio between maximum and minimum values of two vectors $\vec{A}$ and $\vec{B}$ ($\vec{A}>\vec{B}$) is 1:4. Then the ratio between the magnitudes of two vectors is

• A. 3:2
• B. 5:3
• C. 2:3
• D. 3:5

Hint:

Use the componendo-dividendo rule: if $(A+B)/(A-B) = 4/1$, then $A/B = (4+1)/(4-1) = 5/3$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The maximum resultant of two vectors occurs when they are in the same direction (\(\theta = 0^\circ\)), and the minimum occurs when they are in opposite directions (\(\theta = 180^\circ\)).
Step 2: Key Formula or Approach:
1. \(R_{max} = A + B\).
2. \(R_{min} = A - B\).
3. Given \(\frac{A + B}{A - B} = \frac{4}{1}\).
Step 3: Detailed Explanation:
From the given ratio: \[ \frac{A + B}{A - B} = \frac{4}{1} \] Cross-multiply: \[ A + B = 4(A - B) \] \[ A + B = 4A - 4B \] Rearrange the terms: \[ B + 4B = 4A - A \] \[ 5B = 3A \] \[ \frac{A}{B} = \frac{5}{3} \]
Step 4: Final Answer:
The ratio between the magnitudes of the two vectors is 5:3.