Question

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

• A. \(3:2\)
• B. \(5:3\)
• C. \(2:3\)
• D. \(3:5\)

Hint:

For two vectors, maximum resultant is \(A+B\) and minimum resultant is \(A-B\) when \(A>B\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The maximum resultant of two vectors occurs when they are in the same direction (\(\theta = 0^\circ\)).
The minimum resultant occurs when they are in opposite directions (\(\theta = 180^\circ\)).
We are given the ratio of these resultants and need to find the ratio of the individual vector magnitudes.
Step 2: Key Formula or Approach:
Maximum resultant \(R_{\text{max}} = A + B\).
Minimum resultant \(R_{\text{min}} = A - B\).
Given \(R_{\text{max}} : R_{\text{min}} = 4 : 1\).
Solve the linear equation for the ratio \(A/B\).
Step 3: Detailed Explanation:

Let the magnitudes be \(A\) and \(B\) with \(A>B\).

According to the problem statement:
\[ \frac{A + B}{A - B} = \frac{4}{1} \]

Cross-multiplying to solve:
\[ A + B = 4(A - B) \]
\[ A + B = 4A - 4B \]

Grouping \(A\) terms and \(B\) terms:
\[ B + 4B = 4A - A \]
\[ 5B = 3A \]

Therefore, the ratio of magnitudes \(A\) to \(B\) is:
\[ \frac{A}{B} = \frac{5}{3} \]

So, \(A:B = 5:3\).

Step 4: Final Answer:
The ratio between the magnitudes of the two vectors, determined from the ratio of their maximum and minimum resultants, is 5:3.