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.