The velocity of a particle is `v=6hati+2hatj-2hatk` The component of the velocity parallel to vector `a=hati+hatj+2hatk` invector 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 \( \mathbf{v} = 6\hat{i} + 2\hat{j} - 2\hat{k} \) that is parallel to the vector \( \mathbf{a} = \hat{i} + \hat{j} + 2\hat{k} \), we can follow these steps:

### Step 1: Calculate the dot product of \( \mathbf{v} \) and \( \mathbf{a} \)

The dot product \( \mathbf{v} \cdot \mathbf{a} \) is calculated as follows:

\[ \mathbf{v} \cdot \mathbf{a} = (6\hat{i} + 2\hat{j} - 2\hat{k}) \cdot (\hat{i} + \hat{j} + 2\hat{k}) \]

Calculating the dot product:

\[ = 6 \cdot 1 + 2 \cdot 1 + (-2) \cdot 2 \]
\[ = 6 + 2 - 4 = 4 \]

### Step 2: Calculate the magnitude of vector \( \mathbf{a} \)

The magnitude of vector \( \mathbf{a} \) is given by:

\[ |\mathbf{a}| = \sqrt{(1^2 + 1^2 + 2^2)} = \sqrt{1 + 1 + 4} = \sqrt{6} \]

### Step 3: Calculate the component of \( \mathbf{v} \) parallel to \( \mathbf{a} \)

The component of \( \mathbf{v} \) parallel to \( \mathbf{a} \) can be calculated using the formula:

\[ \text{Component of } \mathbf{v} \text{ parallel to } \mathbf{a} = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{a}|^2} \mathbf{a} \]

First, we need \( |\mathbf{a}|^2 \):

\[ |\mathbf{a}|^2 = 6 \]

Now substituting the values:

\[ \text{Component of } \mathbf{v} \text{ parallel to } \mathbf{a} = \frac{4}{6} \mathbf{a} = \frac{2}{3} \mathbf{a} \]

Substituting \( \mathbf{a} \):

\[ = \frac{2}{3} (\hat{i} + \hat{j} + 2\hat{k}) = \frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{4}{3}\hat{k} \]

### Final Answer

Thus, the component of the velocity \( \mathbf{v} \) parallel to the vector \( \mathbf{a} \) is:

\[ \frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{4}{3}\hat{k} \]

---