Three vectors are given below
`vec(a)=hat(i)+2hat(i)+3hat(k),vec(b) = 2hat(i)+4hat(j) + 6 hat(k) and vec(c ) = 3hat(i) + 6hat(j) + 9hat(k)`.
Find the components of the vector `bara+barb-barc`.

Answer

To solve the problem, we need to find the components of the vector \( \vec{a} + \vec{b} - \vec{c} \). Let's break it down step by step.

### Step 1: Write down the vectors
We have the following vectors:
- \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \)
- \( \vec{b} = 2\hat{i} + 4\hat{j} + 6\hat{k} \)
- \( \vec{c} = 3\hat{i} + 6\hat{j} + 9\hat{k} \)

### Step 2: Add vectors \( \vec{a} \) and \( \vec{b} \)
Now, we need to add \( \vec{a} \) and \( \vec{b} \):
\[ \vec{a} + \vec{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (2\hat{i} + 4\hat{j} + 6\hat{k}) \]

### Step 3: Combine like terms
Combine the components of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):
\[ \vec{a} + \vec{b} = (1 + 2)\hat{i} + (2 + 4)\hat{j} + (3 + 6)\hat{k} = 3\hat{i} + 6\hat{j} + 9\hat{k} \]

### Step 4: Subtract vector \( \vec{c} \)
Now, we subtract \( \vec{c} \) from \( \vec{a} + \vec{b} \):
\[ \vec{a} + \vec{b} - \vec{c} = (3\hat{i} + 6\hat{j} + 9\hat{k}) - (3\hat{i} + 6\hat{j} + 9\hat{k}) \]

### Step 5: Combine like terms again
Combine the components:
\[ \vec{a} + \vec{b} - \vec{c} = (3 - 3)\hat{i} + (6 - 6)\hat{j} + (9 - 9)\hat{k} = 0\hat{i} + 0\hat{j} + 0\hat{k} \]

### Step 6: Write the final result
Thus, the components of the vector \( \vec{a} + \vec{b} - \vec{c} \) are:
\[ \vec{a} + \vec{b} - \vec{c} = 0\hat{i} + 0\hat{j} + 0\hat{k} \]

### Conclusion
The components of the vector \( \vec{a} + \vec{b} - \vec{c} \) are \( (0, 0, 0) \).

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