A body, under the action of a force `vec(F)=6hati -8hatj+10hatk` , acquires an acceleration of `1ms^(-2)` . The mass of this body must be.

Options

A. 200 kg
B. 20kg
C. `10sqrt(2) kg`
D. 6sqrt(2) kg`

Explanation

To find the mass of the body under the action of the given force, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. The formula can be expressed as:

\[ \vec{F} = m \cdot \vec{a} \]

Where:
- \( \vec{F} \) is the force vector,
- \( m \) is the mass of the body,
- \( \vec{a} \) is the acceleration vector.

Given:
- Force \( \vec{F} = 6 \hat{i} - 8 \hat{j} + 10 \hat{k} \)
- Acceleration \( \vec{a} = 1 \, \text{m/s}^2 \)

### Step 1: Calculate the magnitude of the force vector

The magnitude of the force vector \( \vec{F} \) can be calculated using the formula:

\[ |\vec{F}| = \sqrt{(F_x)^2 + (F_y)^2 + (F_z)^2} \]

Where \( F_x, F_y, F_z \) are the components of the force vector.

Substituting the values:

\[ |\vec{F}| = \sqrt{(6)^2 + (-8)^2 + (10)^2} \]
\[ |\vec{F}| = \sqrt{36 + 64 + 100} \]
\[ |\vec{F}| = \sqrt{200} \]

### Step 2: Calculate the mass using Newton's second law

From Newton's second law, we can rearrange the equation to find mass:

\[ m = \frac{|\vec{F}|}{|\vec{a}|} \]

Since the acceleration \( \vec{a} \) is given as \( 1 \, \text{m/s}^2 \), we have:

\[ m = \frac{\sqrt{200}}{1} \]
\[ m = \sqrt{200} \]

### Step 3: Simplify the mass

Now, we can simplify \( \sqrt{200} \):

\[ \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \]

### Conclusion

Thus, the mass of the body is:

\[ m = 10\sqrt{2} \, \text{kg} \]

### Final Answer

The mass of the body must be \( 10\sqrt{2} \) kg, which corresponds to option 3.

---