The component of `hati` in the direction of vector `hati+hatj+2hatk` is

Options

A. 1
B. 2
C. `sqrt6`
D. `1/sqrt6`

Answer

To find the component of the vector \(\hat{i}\) in the direction of the vector \(\hat{i} + \hat{j} + 2\hat{k}\), we can follow these steps:

### Step 1: Identify the vectors
Let:
- Vector \( \mathbf{a} = \hat{i} \)
- Vector \( \mathbf{b} = \hat{i} + \hat{j} + 2\hat{k} \)

### Step 2: Find the magnitude of vector \( \mathbf{b} \)
The magnitude of vector \( \mathbf{b} \) is calculated as follows:
\[ |\mathbf{b}| = \sqrt{(1)^2 + (1)^2 + (2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]

### Step 3: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \)
Now, we compute the dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{a} \cdot \mathbf{b} = \hat{i} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = \hat{i} \cdot \hat{i} + \hat{i} \cdot \hat{j} + \hat{i} \cdot (2\hat{k}) = 1 + 0 + 0 = 1 \]

### Step 4: Find the component of \( \mathbf{a} \) in the direction of \( \mathbf{b} \)
The component of vector \( \mathbf{a} \) in the direction of vector \( \mathbf{b} \) is given by the formula:
\[ \text{Component of } \mathbf{a} \text{ in the direction of } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]
Substituting the values we found:
\[ \text{Component} = \frac{1}{\sqrt{6}} \]

### Final Answer
The component of \( \hat{i} \) in the direction of \( \hat{i} + \hat{j} + 2\hat{k} \) is \( \frac{1}{\sqrt{6}} \).

---