Vector sum of two forces of 10N and 6N cannot be:

Options

A. 4N
B. 8N
C. 12N
D. 2N

Answer

To solve the problem of finding the vector sum of two forces of 10N and 6N, we will follow these steps:

### Step 1: Understand the Concept of Vector Sum
The vector sum of two forces can be calculated using the formula for the resultant of two vectors:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
where:
- \( R \) is the resultant vector,
- \( A \) and \( B \) are the magnitudes of the two vectors,
- \( \theta \) is the angle between the two vectors.

### Step 2: Identify the Forces
Here, we have:
- Force \( A = 10N \)
- Force \( B = 6N \)

### Step 3: Determine the Minimum and Maximum Values of the Resultant
1. **Minimum Value of R**: This occurs when the two vectors are in opposite directions (i.e., \( \theta = 180^\circ \), where \( \cos 180^\circ = -1 \)):
\[ R_{\text{min}} = |A - B| = |10N - 6N| = 4N \]

2. **Maximum Value of R**: This occurs when the two vectors are in the same direction (i.e., \( \theta = 0^\circ \), where \( \cos 0^\circ = 1 \)):
\[ R_{\text{max}} = A + B = 10N + 6N = 16N \]

### Step 4: Establish the Range of Possible Values
From the calculations above, the resultant vector \( R \) can take values in the range:
\[ 4N \leq R \leq 16N \]

### Step 5: Analyze the Given Options
The question asks which value cannot be the vector sum of the two forces. The possible values given are:
- 4N
- 8N
- 12N
- 2N

### Step 6: Determine the Impossible Value
Since the minimum resultant is 4N and the maximum is 16N, any resultant value must lie within this range. Therefore, the only value that cannot be achieved is:
- **2N** (as it is less than the minimum value of 4N).

### Final Answer
The vector sum of two forces of 10N and 6N cannot be **2N**.

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