Find the sum of the vectors : `vec(a)=hat(i)-2hat(j),vec(b)=-2hat(i)-3hat(j)` and `vec(c )=2hat(i)+3hat(k)`.

Answer

To find the sum of the vectors \(\vec{a} = \hat{i} - 2\hat{j}\), \(\vec{b} = -2\hat{i} - 3\hat{j}\), and \(\vec{c} = 2\hat{i} + 3\hat{k}\), we will add the corresponding components of each vector.

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

### Step 2: Group the components
We will group the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components separately.

- For \( \hat{i} \) components:
\[ \text{Sum of } \hat{i} = 1 + (-2) + 2 \]

- For \( \hat{j} \) components:
\[ \text{Sum of } \hat{j} = -2 + (-3) + 0 \]

- For \( \hat{k} \) components:
\[ \text{Sum of } \hat{k} = 0 + 0 + 3 \]

### Step 3: Calculate the sums
- Calculate the \( \hat{i} \) sum:
\[ 1 - 2 + 2 = 1 \]

- Calculate the \( \hat{j} \) sum:
\[ -2 - 3 = -5 \]

- Calculate the \( \hat{k} \) sum:
\[ 0 + 0 + 3 = 3 \]

### Step 4: Combine the results
Now, we combine the sums of the components:
\[ \vec{S} = 1\hat{i} - 5\hat{j} + 3\hat{k} \]

### Final Result
Thus, the sum of the vectors is:
\[ \vec{S} = \hat{i} - 5\hat{j} + 3\hat{k} \]

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