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 find the components of the vector \(\vec{a} + \vec{b} - \vec{c}\), we will follow these steps:

### Step 1: Write down the given vectors
The vectors are given as:
\[ \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \]
\[ \vec{b} = 2\hat{i} + 4\hat{j} + 6\hat{k} \]
\[ \vec{c} = 3\hat{i} + 6\hat{j} + 9\hat{k} \]

### Step 2: Add vectors \(\vec{a}\) and \(\vec{b}\)
We will add the components of \(\vec{a}\) and \(\vec{b}\):
\[ \vec{a} + \vec{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (2\hat{i} + 4\hat{j} + 6\hat{k}) \]
Combine the components:
\[ = (1 + 2)\hat{i} + (2 + 4)\hat{j} + (3 + 6)\hat{k} \]
\[ = 3\hat{i} + 6\hat{j} + 9\hat{k} \]

### Step 3: Subtract vector \(\vec{c}\)
Now, we will subtract \(\vec{c}\) from the result of \(\vec{a} + \vec{b}\):
\[ \vec{a} + \vec{b} - \vec{c} = (3\hat{i} + 6\hat{j} + 9\hat{k}) - (3\hat{i} + 6\hat{j} + 9\hat{k}) \]
Combine the components:
\[ = (3 - 3)\hat{i} + (6 - 6)\hat{j} + (9 - 9)\hat{k} \]
\[ = 0\hat{i} + 0\hat{j} + 0\hat{k} \]

### Step 4: Final Result
Thus, the components of the vector \(\vec{a} + \vec{b} - \vec{c}\) are:
\[ \vec{a} + \vec{b} - \vec{c} = \vec{0} \]

---