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 use the formula for the component of one vector in the direction of another vector. The formula is given by:

\[ \text{Component of } \mathbf{a} \text{ in the direction of } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]

### Step-by-step Solution:

1. **Identify Vectors**:
- Let \( \mathbf{a} = \hat{i} \)
- Let \( \mathbf{b} = \hat{i} + \hat{j} + 2\hat{k} \)

2. **Calculate the Dot Product \( \mathbf{a} \cdot \mathbf{b} \)**:
- The dot product \( \mathbf{a} \cdot \mathbf{b} = \hat{i} \cdot (\hat{i} + \hat{j} + 2\hat{k}) \)
- Using the properties of dot products:
\[ \hat{i} \cdot \hat{i} = 1, \quad \hat{i} \cdot \hat{j} = 0, \quad \hat{i} \cdot \hat{k} = 0 \]
- Therefore:
\[ \mathbf{a} \cdot \mathbf{b} = 1 + 0 + 0 = 1 \]

3. **Calculate the Magnitude of \( \mathbf{b} \)**:
- The magnitude \( |\mathbf{b}| = |\hat{i} + \hat{j} + 2\hat{k}| \)
- This can be calculated as:
\[ |\mathbf{b}| = \sqrt{(1^2 + 1^2 + 2^2)} = \sqrt{1 + 1 + 4} = \sqrt{6} \]

4. **Calculate the Component**:
- Now, substitute the values into the component formula:
\[ \text{Component of } \hat{i} \text{ in the direction of } \hat{i} + \hat{j} + 2\hat{k} = \frac{1}{\sqrt{6}} \]

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