If `a=3hati-2hatj+hatk,b=2hati-4hatj-3hatk and c=-hati+2hatj+2hatk`, then a+b+c is

Options

A. `3hati-4hatj`
B. `3hati+4hatj`
C. `4hati-4hatj`
D. `4hati+4hatj`

Explanation

To solve the problem, we need to find the sum of the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \).

Given:
- \( \mathbf{a} = 3\hat{i} - 2\hat{j} + \hat{k} \)
- \( \mathbf{b} = 2\hat{i} - 4\hat{j} - 3\hat{k} \)
- \( \mathbf{c} = -\hat{i} + 2\hat{j} + 2\hat{k} \)

### Step 1: Write the expression for \( \mathbf{a} + \mathbf{b} + \mathbf{c} \)

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

### Step 2: Combine the \( \hat{i} \) components

\[ \text{Total } \hat{i} = 3 + 2 - 1 = 4 \]

### Step 3: Combine the \( \hat{j} \) components

\[ \text{Total } \hat{j} = -2 - 4 + 2 = -4 \]

### Step 4: Combine the \( \hat{k} \) components

\[ \text{Total } \hat{k} = 1 - 3 + 2 = 0 \]

### Step 5: Write the final result

Thus, the sum of the vectors \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is:

\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = 4\hat{i} - 4\hat{j} + 0\hat{k} \]

This can be simplified to:

\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = 4\hat{i} - 4\hat{j} \]

### Final Answer:
\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = 4\hat{i} - 4\hat{j} \]

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