Statement1: A component of vector `vecb = 4hati + 2hatj + 3hatk` in the direction perpendicular to the direction of vector `veca = hati + hatj +hatk is hati - hatj`
Statement 2: A component of vector in the direction of `veca = hati + hatj + hatk is 2hati + 2hatj + 2hatk`

Options

A. (a) Both the statements are true and statement 2 is the correct explanation for statement 1.
B. (b) Both statements are true but statement 2 is not the correct explanation for statement 1.
C. (c) Statement 1 is true and Statement 2 is false
D. (d)Statement 1 is false and Statement 2 is true.

Explanation

To solve the problem, we need to analyze both statements regarding the vectors given.

### Given:
- Vector **b** = \(4\hat{i} + 2\hat{j} + 3\hat{k}\)
- Vector **a** = \(\hat{i} + \hat{j} + \hat{k}\)
- Perpendicular direction to **a** = \(\hat{i} - \hat{j}\)

### Step 1: Find the unit vector in the direction of vector **a**.
The unit vector **a** (denoted as **â**) can be calculated as follows:

\[ \text{Magnitude of } \vec{a} = |\vec{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

\[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) \]

### Step 2: Find the component of vector **b** in the direction of vector **a**.
The component of vector **b** in the direction of **a** can be calculated using the formula:

\[ \text{Component of } \vec{b} \text{ in the direction of } \vec{a} = \left(\vec{b} \cdot \hat{a}\right) \hat{a} \]

Calculating the dot product \(\vec{b} \cdot \hat{a}\):

\[ \vec{b} \cdot \hat{a} = (4\hat{i} + 2\hat{j} + 3\hat{k}) \cdot \left(\frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})\right) \]

\[ = \frac{1}{\sqrt{3}}(4 \cdot 1 + 2 \cdot 1 + 3 \cdot 1) = \frac{1}{\sqrt{3}}(4 + 2 + 3) = \frac{9}{\sqrt{3}} = 3\sqrt{3} \]

Now, the component of **b** in the direction of **a** is:

\[ \text{Component of } \vec{b} \text{ in the direction of } \vec{a} = 3\sqrt{3} \cdot \hat{a} = 3\sqrt{3} \cdot \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) = 3(\hat{i} + \hat{j} + \hat{k}) = 3\hat{i} + 3\hat{j} + 3\hat{k} \]

### Step 3: Analyze Statement 2
Statement 2 claims that the component of vector **b** in the direction of vector **a** is \(2\hat{i} + 2\hat{j} + 2\hat{k}\). We found it to be \(3\hat{i} + 3\hat{j} + 3\hat{k}\), so Statement 2 is **false**.

### Step 4: Find the component of vector **b** in the direction perpendicular to vector **a**.
To find the component of **b** in the direction perpendicular to **a**, we can use the formula:

\[ \text{Component of } \vec{b} \text{ perpendicular to } \vec{a} = \vec{b} - \text{Component of } \vec{b} \text{ in the direction of } \vec{a} \]

Calculating this:

\[ \text{Component of } \vec{b} \text{ perpendicular to } \vec{a} = \vec{b} - (3\hat{i} + 3\hat{j} + 3\hat{k}) \]

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

\[ = 1\hat{i} - 1\hat{j} + 0\hat{k} = \hat{i} - \hat{j} \]

### Step 5: Analyze Statement 1
Statement 1 claims that the component of vector **b** in the direction perpendicular to vector **a** is \(\hat{i} - \hat{j}\). We found it to be \(\hat{i} - \hat{j}\), so Statement 1 is **true**.

### Conclusion
- **Statement 1**: True
- **Statement 2**: False

### Final Answer
Statement 1 is true, and Statement 2 is false.

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