Suppose `vecv = vec2i + vecj - veck` and vecw = `veci + vec3k`

Options

A. 81
B. 78
C. 57
D. 54

Answer

To solve the question regarding the vectors \(\vec{v} = 2\vec{i} + \vec{j} - \vec{k}\) and \(\vec{w} = \vec{i} + 3\vec{k}\), we can perform operations such as addition, subtraction, or finding the dot product or cross product. Here, we will find the sum of the two vectors.

### Step-by-Step Solution:

1. **Write down the vectors:**
\[ \vec{v} = 2\vec{i} + \vec{j} - \vec{k} \]
\[ \vec{w} = \vec{i} + 3\vec{k} \]

2. **Add the vectors:**
To add the vectors, we combine the corresponding components (i.e., the coefficients of \(\vec{i}\), \(\vec{j}\), and \(\vec{k}\)).
\[ \vec{v} + \vec{w} = (2\vec{i} + \vec{j} - \vec{k}) + (\vec{i} + 3\vec{k}) \]

3. **Combine the \(\vec{i}\) components:**
\[ 2\vec{i} + \vec{i} = 3\vec{i} \]

4. **Combine the \(\vec{j}\) components:**
\[ \vec{j} + 0 = \vec{j} \]

5. **Combine the \(\vec{k}\) components:**
\[ -\vec{k} + 3\vec{k} = 2\vec{k} \]

6. **Write the resultant vector:**
Combining all the components together, we get:
\[ \vec{v} + \vec{w} = 3\vec{i} + \vec{j} + 2\vec{k} \]

### Final Answer:
\[ \vec{v} + \vec{w} = 3\vec{i} + \vec{j} + 2\vec{k} \]