Which one of the following pair cannot be the rectangular components of force vector of 10 N ?

Options

A. `6 N & 8 N `
B. `7 N & sqrt(51) N`
C. `6sqrt(2) N & 2 sqrt(7) N`
D. `9 N & 1 N`

Answer

To determine which pair cannot be the rectangular components of a force vector of 10 N, we need to use the Pythagorean theorem, which states that for two perpendicular components \( F_x \) and \( F_y \) of a vector \( F \), the relationship is given by:

\[ F^2 = F_x^2 + F_y^2 \]

Given that \( F = 10 \, \text{N} \), we have:

\[ 10^2 = F_x^2 + F_y^2 \]

This simplifies to:

\[ 100 = F_x^2 + F_y^2 \]

Now, we will evaluate each of the given pairs to see if they satisfy this equation.

### Step 1: Evaluate the first pair (6 N, 8 N)

\[ F_x = 6 \, \text{N}, \quad F_y = 8 \, \text{N} \]

Calculating:

\[ F_x^2 + F_y^2 = 6^2 + 8^2 = 36 + 64 = 100 \]

This pair satisfies the equation.

### Step 2: Evaluate the second pair (7 N, \( \sqrt{51} \) N)

\[ F_x = 7 \, \text{N}, \quad F_y = \sqrt{51} \, \text{N} \]

Calculating:

\[ F_x^2 + F_y^2 = 7^2 + (\sqrt{51})^2 = 49 + 51 = 100 \]

This pair also satisfies the equation.

### Step 3: Evaluate the third pair (6√2 N, 2√7 N)

\[ F_x = 6\sqrt{2} \, \text{N}, \quad F_y = 2\sqrt{7} \, \text{N} \]

Calculating:

\[ F_x^2 + F_y^2 = (6\sqrt{2})^2 + (2\sqrt{7})^2 = 72 + 28 = 100 \]

This pair satisfies the equation as well.

### Step 4: Evaluate the fourth pair (9 N, 1 N)

\[ F_x = 9 \, \text{N}, \quad F_y = 1 \, \text{N} \]

Calculating:

\[ F_x^2 + F_y^2 = 9^2 + 1^2 = 81 + 1 = 82 \]

This pair does not satisfy the equation.

### Conclusion

The pair that cannot be the rectangular components of a force vector of 10 N is:

**Option 4: (9 N, 1 N)**

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