Given the vectors $\mathbf{a} = 4\mathbf{i} - 11\mathbf{j} + 17\mathbf{k}$, $\mathbf{b} = \mathbf{i} + 4\mathbf{j} + 6\mathbf{k}$, and $\mathbf{c} = -\mathbf{i} + 3\mathbf{j} - 5\mathbf{k}$, we need to calculate the vector expression $\mathbf{a} - \mathbf{b} + \mathbf{c}$.

Step-by-Step Solution

1. Subtract $\mathbf{b}$ from $\mathbf{a}$:

   $\mathbf{a} - \mathbf{b} = \left(4\mathbf{i} - 11\mathbf{j} + 17\mathbf{k}\right) - \left(\mathbf{i} + 4\mathbf{j} + 6\mathbf{k}\right)$

   Subtracting the corresponding components:

   $= \left(4 - 1\right)\mathbf{i} + \left(-11 - 4\right)\mathbf{j} + \left(17 - 6\right)\mathbf{k}$

   $= 3\mathbf{i} - 15\mathbf{j} + 11\mathbf{k}$

2. Add $\mathbf{c}$ to the result:

   $\mathbf{a} - \mathbf{b} + \mathbf{c} = \left(3\mathbf{i} - 15\mathbf{j} + 11\mathbf{k}\right) + \left(-\mathbf{i} + 3\mathbf{j} - 5\mathbf{k}\right)$

   Adding the corresponding components:

   $= \left(3 - 1\right)\mathbf{i} + \left(-15 + 3\right)\mathbf{j} + \left(11 - 5\right)\mathbf{k}$

   $= 2\mathbf{i} - 12\mathbf{j} + 6\mathbf{k}$

Final Answer

$\mathbf{a} - \mathbf{b} + \mathbf{c} = 2\mathbf{i} - 12\mathbf{j} + 6\mathbf{k}$

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the magnitude of the resulting vector $\mathbf{a} - \mathbf{b} + \mathbf{c}$?
2. How would you find the direction angles of the vector $\mathbf{a} - \mathbf{b} + \mathbf{c}$?
3. If $\mathbf{a}$ and $\mathbf{b}$ were perpendicular, what would their dot product be?
4. How would you compute the cross product of $\mathbf{a}$ and $\mathbf{b}$?
5. How does vector subtraction differ from scalar subtraction in terms of operations?

Tip: When working with vectors, always align corresponding components when adding or subtracting to ensure accuracy.