Let vectors A⃗ =(2,−1,1)A→=(2,−1,1), B⃗ =(3,0,5)B→=(3,0,5), and C⃗ =(1,4,−2)C→=(1,4,−2), where (x,y,z)(x,y,z) are the components of the vectors along i^i^, j^j^, and k^k^ respectively. Calculate the following: **Express your answer as an ordered triplet of components** (x,y,z)(x,y,z) **with commas to separate the components.**

To solve this problem, you need to clarify what specific calculation you want to perform with the vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\). Here are some possible calculations:

1. **Vector addition:** \(\mathbf{A} + \mathbf{B}\) or \(\mathbf{A} + \mathbf{B} + \mathbf{C}\)
2. **Vector subtraction:** \(\mathbf{A} - \mathbf{B}\), \(\mathbf{A} - \mathbf{C}\), etc.
3. **Dot product:** \(\mathbf{A} \cdot \mathbf{B}\), \(\mathbf{A} \cdot \mathbf{C}\), etc.
4. **Cross product:** \(\mathbf{A} \times \mathbf{B}\), \(\mathbf{A} \times \mathbf{C}\), etc.

Could you please specify which operation you'd like to perform?

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**Relative Questions:**
1. How would you calculate the sum of two vectors?
2. What is the geometric interpretation of the dot product of two vectors?
3. How does the cross product of two vectors differ from their dot product?
4. Can you explain how vector subtraction works in terms of component-wise operations?
5. What does the magnitude of a vector represent in physical terms?

**Tip:** When dealing with vector operations, it's useful to break them down component-wise to simplify the calculations.

Learn how to calculate the components of vectors A, B, and C using vector addition, subtraction, dot product, and cross product operations. Understand the mathematical concepts involved and get detailed solutions. Suitable for undergraduate-level studies.

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