Question

Among the following pairs of vectors, if the resultant of two vectors can never have magnitude 4 units, the magnitudes of the vectors are

• A. 2 units and 2 units
• B. 1 unit and 3 units
• C. 5 units and 1 unit
• D. 7 units and 2 units
• E. 5 units and 8 units

Hint:

Resultant magnitude always lies between sum and difference of magnitudes.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The magnitude of the resultant vector \(\vec{R}\) of two vectors \(\vec{A}\) and \(\vec{B}\) depends on the angle \(\theta\) between them. The resultant magnitude `R` is bounded by a minimum and a maximum value. The maximum occurs when the vectors are parallel (\(\theta=0^\circ\)), and the minimum occurs when they are anti-parallel (\(\theta=180^\circ\)).
Step 2: Key Formula or Approach:
For two vectors with magnitudes A and B, the magnitude of their resultant, R, must lie in the range: \[ |A - B| \le R \le A + B \] We need to check this condition for each pair of magnitudes to see which range does not include the value 4.
Step 3: Detailed Explanation:
Let's check the range of possible resultant magnitudes for each option:
(A) 2 units and 2 units: Maximum resultant: \(2 + 2 = 4\). Minimum resultant: \(|2 - 2| = 0\). Range: \([0, 4]\). The resultant can be 4.
(B) 1 unit and 3 units: Maximum resultant: \(1 + 3 = 4\). Minimum resultant: \(|1 - 3| = 2\). Range: \([2, 4]\). The resultant can be 4.
(C) 5 units and 1 unit: Maximum resultant: \(5 + 1 = 6\). Minimum resultant: \(|5 - 1| = 4\). Range: \([4, 6]\). The resultant can be 4.
(D) 7 units and 2 units: Maximum resultant: \(7 + 2 = 9\). Minimum resultant: \(|7 - 2| = 5\). Range: \([5, 9]\). The value 4 is outside this range. Therefore, the resultant can never be 4.
(E) 5 units and 8 units: Maximum resultant: \(5 + 8 = 13\). Minimum resultant: \(|5 - 8| = 3\). Range: \([3, 13]\). The resultant can be 4.
Step 4: Final Answer:
The pair of vectors with magnitudes 7 units and 2 units can never have a resultant of magnitude 4.