The velocity of a particle is `v=6hati+2hatj-2hatk` The component of the velocity parallel to vector `a=hati+hatj+hatk` in vector from is

Options

A. `6hati +2hatj +2hatk`
B. `2hati +2hatj +2hatk`
C. `hati +hatj +hatk`
D. `6hati +2hatj +2hatk`

Explanation

To find the component of the velocity vector \( \mathbf{v} = 6\hat{i} + 2\hat{j} - 2\hat{k} \) parallel to the vector \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \), we can follow these steps:

### Step 1: Calculate the magnitude of vector \( \mathbf{a} \)
The magnitude of vector \( \mathbf{a} \) is given by:
\[ |\mathbf{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

### Step 2: Calculate the dot product \( \mathbf{v} \cdot \mathbf{a} \)
Now, we calculate the dot product of \( \mathbf{v} \) and \( \mathbf{a} \):
\[ \mathbf{v} \cdot \mathbf{a} = (6\hat{i} + 2\hat{j} - 2\hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) \]
Calculating this gives:
\[ = 6 \cdot 1 + 2 \cdot 1 + (-2) \cdot 1 = 6 + 2 - 2 = 6 \]

### Step 3: Calculate the component of \( \mathbf{v} \) along \( \mathbf{a} \)
The magnitude of the component of \( \mathbf{v} \) along \( \mathbf{a} \) is given by:
\[ \text{Component of } \mathbf{v} \text{ along } \mathbf{a} = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{a}|} \]
Substituting the values we calculated:
\[ = \frac{6}{\sqrt{3}} = 2\sqrt{3} \]

### Step 4: Find the unit vector of \( \mathbf{a} \)
The unit vector \( \hat{a} \) in the direction of \( \mathbf{a} \) is:
\[ \hat{a} = \frac{\mathbf{a}}{|\mathbf{a}|} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \]

### Step 5: Calculate the component of \( \mathbf{v} \) in vector form
The component of \( \mathbf{v} \) parallel to \( \mathbf{a} \) in vector form is:
\[ \text{Component of } \mathbf{v} \text{ along } \mathbf{a} = \left(2\sqrt{3}\right) \hat{a} \]
Substituting \( \hat{a} \):
\[ = 2\sqrt{3} \cdot \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} = 2(\hat{i} + \hat{j} + \hat{k}) = 2\hat{i} + 2\hat{j} + 2\hat{k} \]

### Final Answer
Thus, the component of the velocity parallel to vector \( \mathbf{a} \) in vector form is:
\[ \mathbf{v}_{\parallel} = 2\hat{i} + 2\hat{j} + 2\hat{k} \]