If `vec(A)=2hati+hatj+hatk " and " vecB=10hati+5hatj+5hatk`, if the magnitude of component of `(vec(B)-vec(A))` along `vec(A)` is `4sqrt(x)`. Then x will be .

Explanation

To solve the problem, we need to find the value of \( x \) given the vectors \( \vec{A} \) and \( \vec{B} \), and the condition regarding the magnitude of the component of \( \vec{B} - \vec{A} \) along \( \vec{A} \).

### Step-by-step Solution:

1. **Define the vectors**:
\[ \vec{A} = 2\hat{i} + \hat{j} + \hat{k} \]
\[ \vec{B} = 10\hat{i} + 5\hat{j} + 5\hat{k} \]

2. **Calculate \( \vec{B} - \vec{A} \)**:
\[ \vec{B} - \vec{A} = (10\hat{i} + 5\hat{j} + 5\hat{k}) - (2\hat{i} + \hat{j} + \hat{k}) \]
\[ = (10 - 2)\hat{i} + (5 - 1)\hat{j} + (5 - 1)\hat{k} \]
\[ = 8\hat{i} + 4\hat{j} + 4\hat{k} \]

3. **Factor out the common term**:
\[ \vec{B} - \vec{A} = 4(2\hat{i} + \hat{j} + \hat{k}) \]

4. **Let \( \vec{R} = \vec{B} - \vec{A} = 4(2\hat{i} + \hat{j} + \hat{k}) \)**.

5. **Find the magnitude of \( \vec{A} \)**:
\[ |\vec{A}| = \sqrt{(2)^2 + (1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

6. **Calculate the dot product \( \vec{R} \cdot \vec{A} \)**:
\[ \vec{R} \cdot \vec{A} = (4(2\hat{i} + \hat{j} + \hat{k})) \cdot (2\hat{i} + \hat{j} + \hat{k}) \]
\[ = 4[(2)(2) + (1)(1) + (1)(1)] = 4[4 + 1 + 1] = 4 \times 6 = 24 \]

7. **Find the component of \( \vec{R} \) along \( \vec{A} \)**:
\[ \text{Component of } \vec{R} \text{ along } \vec{A} = \frac{\vec{R} \cdot \vec{A}}{|\vec{A}|} \]
\[ = \frac{24}{\sqrt{6}} \]

8. **Set the component equal to \( 4\sqrt{x} \)**:
\[ \frac{24}{\sqrt{6}} = 4\sqrt{x} \]

9. **Simplify the equation**:
\[ 24 = 4\sqrt{x} \cdot \sqrt{6} \]
\[ 24 = 4\sqrt{6x} \]
\[ 6 = \sqrt{6x} \]

10. **Square both sides**:
\[ 36 = 6x \]
\[ x = 6 \]

### Final Answer:
\[ x = 6 \]