Let's calculate the vector operations one by one for vectors $\mathbf{A} = \langle 3, 4, 5 \rangle$ and $\mathbf{B} = \langle -3, 5, -6 \rangle$.

1. $\mathbf{A} + \mathbf{B}$

To add two vectors, add their corresponding components:

$\mathbf{A} + \mathbf{B} = \langle 3 + (-3), 4 + 5, 5 + (-6) \rangle$

$\mathbf{A} + \mathbf{B} = \langle 0, 9, -1 \rangle$

2. $\mathbf{A} - \mathbf{B}$

To subtract vector $\mathbf{B}$ from vector $\mathbf{A}$, subtract their corresponding components:

$\mathbf{A} - \mathbf{B} = \langle 3 - (-3), 4 - 5, 5 - (-6) \rangle$

$\mathbf{A} - \mathbf{B} = \langle 6, -1, 11 \rangle$

3. $\mathbf{B} - \mathbf{A}$

To subtract vector $\mathbf{A}$ from vector $\mathbf{B}$, subtract their corresponding components:

$\mathbf{B} - \mathbf{A} = \langle -3 - 3, 5 - 4, -6 - 5 \rangle$

$\mathbf{B} - \mathbf{A} = \langle -6, 1, -11 \rangle$

Summary of results:

• $\mathbf{A} + \mathbf{B} = \langle 0, 9, -1 \rangle$
• $\mathbf{A} - \mathbf{B} = \langle 6, -1, 11 \rangle$
• $\mathbf{B} - \mathbf{A} = \langle -6, 1, -11 \rangle$

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the magnitude of vector $\mathbf{A}$?
2. How do you calculate the dot product of two vectors?
3. What is the result of the cross product $\mathbf{A} \times \mathbf{B}$?
4. How can you find the angle between vectors $\mathbf{A}$ and $\mathbf{B}$?
5. What is the projection of vector $\mathbf{A}$ onto vector $\mathbf{B}$?

Tip: When performing vector operations, always ensure you are matching the correct components to avoid sign errors in your calculations.