It is supposed to be addition and subtraction 

Thank you for clarifying. Let's redo the calculations based on the operations of addition and subtraction instead of dot and cross products.

### Given:
1. **\(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\)**
2. **\(\vec{b} = -\hat{i} + 3\hat{j} + 5\hat{k}\)**

### Task:
1. **Find \(\vec{a} + \vec{b}\)** (Vector Addition)
2. **Find \(\vec{a} - \vec{b}\)** (Vector Subtraction)

### Step 1: Find \(\vec{a} + \vec{b}\)
To add the vectors \(\vec{a}\) and \(\vec{b}\), you simply add their corresponding components:

\[
\vec{a} + \vec{b} = (3\hat{i} + 2\hat{j} - \hat{k}) + (-\hat{i} + 3\hat{j} + 5\hat{k})
\]

Combine the \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) components:

\[
\vec{a} + \vec{b} = (3 - 1)\hat{i} + (2 + 3)\hat{j} + (-1 + 5)\hat{k}
\]
\[
\vec{a} + \vec{b} = 2\hat{i} + 5\hat{j} + 4\hat{k}
\]

### Step 2: Find \(\vec{a} - \vec{b}\)
To subtract the vectors \(\vec{a}\) and \(\vec{b}\), you subtract their corresponding components:

\[
\vec{a} - \vec{b} = (3\hat{i} + 2\hat{j} - \hat{k}) - (-\hat{i} + 3\hat{j} + 5\hat{k})
\]

Combine the \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) components:

\[
\vec{a} - \vec{b} = (3 - (-1))\hat{i} + (2 - 3)\hat{j} + (-1 - 5)\hat{k}
\]
\[
\vec{a} - \vec{b} = 4\hat{i} - \hat{j} - 6\hat{k}
\]

### Final Answers:
1. **\(\vec{a} + \vec{b} = 2\hat{i} + 5\hat{j} + 4\hat{k}\)**
2. **\(\vec{a} - \vec{b} = 4\hat{i} - \hat{j} - 6\hat{k}\)**

Would you like more details on this solution, or do you have any other questions?

### Related Questions:
1. How can vector addition be visualized geometrically?
2. What are the properties of vector addition?
3. How do you find the magnitude of a vector after addition or subtraction?
4. How does vector subtraction relate to the concept of displacement in physics?
5. Can you perform vector operations in a different coordinate system, such as cylindrical or spherical coordinates?

### Tip:
When adding or subtracting vectors, always make sure to handle each component (i.e., \(\hat{i}\), \(\hat{j}\), \(\hat{k}\)) separately and correctly account for signs.

Explore how to perform vector addition and subtraction step-by-step. This problem involves vectors \(\vec{a}\) and \(\vec{b}\), calculating their sum and difference using component-wise addition and subtraction methods. Suitable for high school students.

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