Question

Among the given pair of vectors, the resultant of two vectors can never be 3 units. The vectors are

• A. 1 unit and 2 units
• B. 2 units and 5 units
• C. 3 units and 6 units
• D. 4 units and 8 units

Answer 1

The Correct Option is D

The magnitude of the resultant \(R\) of two vectors \(A\) and \(B\) must satisfy the following inequality:

\[|A - B| \leq R \leq A + B\]

We need to check for which pair of vectors, the resultant cannot be 3 units.

1. (A) 1 unit and 2 units \[|1 - 2| \leq R \leq 1 + 2\] \[1 \leq R \leq 3\] Here, \(R\) can be 3.
2. (B) 2 units and 5 units \[|2 - 5| \leq R \leq 2 + 5\] \[3 \leq R \leq 7\] Here, \(R\) can be 3.
3. (C) 3 units and 6 units \[|3 - 6| \leq R \leq 3 + 6\] \[3 \leq R \leq 9\] Here, \(R\) can be 3.
4. (D) 4 units and 8 units \[|4 - 8| \leq R \leq 4 + 8\] \[4 \leq R \leq 12\] Here, the minimum value of \(R\) is 4. Therefore, \(R\) can never be 3.

Answer: The vectors are (D) 4 units and 8 units. The resultant can never be 3 units.

Answer 2

The resultant \( R \) of two vectors \( \vec{A} \) and \( \vec{B} \) can be found using the following relation, depending on the angle \( \theta \) between them:

\[R = \sqrt{A^2 + B^2 + 2AB \cos \theta}\]

Where:
- \( A \) and \( B \) are the magnitudes of the two vectors,
- \( \theta \) is the angle between the two vectors.

The minimum resultant occurs when the vectors are opposite to each other (\( \theta = 180^\circ \)), and the maximum resultant occurs when the vectors are in the same direction (\( \theta = 0^\circ \)).

Let's check each pair:

1 unit and 2 units: 
 The maximum resultant is \( R = 1 + 2 = 3 \) units, and the minimum resultant is \( R = |2 - 1| = 1 \) unit. Hence, 3 units is possible.

2 units and 5 units:  
 The maximum resultant is \( R = 2 + 5 = 7 \) units, and the minimum resultant is \( R = |5 - 2| = 3 \) units. Hence, 3 units is possible.

3 units and 6 units:  
 The maximum resultant is \( R = 3 + 6 = 9 \) units, and the minimum resultant is \( R = |6 - 3| = 3 \) units. Hence, 3 units is possible.

4 units and 8 units:  
 The maximum resultant is \( R = 4 + 8 = 12 \) units, and the minimum resultant is \( R = |8 - 4| = 4 \) units. Therefore, 3 units is not possible in this case.

Thus, the pair of vectors where the resultant can never be 3 units is \( 4 \) units and \( 8 \) units.