let a⃗ =4i^−11j^+17k^
, b⃗ =i^+4j^+6k^
and c⃗ =−i^+3j^−5k^
, then a⃗ −b⃗ +c⃗ =

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.

Learn how to calculate the vector expression a⃗ - b⃗ + c⃗ with detailed step-by-step solution. This problem covers vector operations, including vector addition and subtraction. Suitable for students in Grades 11-12.

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