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## Mathematical Solution: Vector Cross Product Problem
### Overview
The image displays a step-by-step mathematical solution to a vector algebra problem. The task is to find the vector **v** with the smallest magnitude that satisfies the cross product equation: `(1, 2, -5) × v = (90, 30, 30)`. The solution process includes an initial incorrect attempt, a verification step that identifies the error, and a corrected final solution.
### Components/Axes
The content is structured as a series of mathematical derivations and text explanations. There are no graphical axes or charts. The primary components are:
1. **Problem Statement:** Presented at the top.
2. **Initial Solution Attempt:** Labeled "Reponse:" (likely a typo for "Response").
3. **Verification Section:** Introduced with green text: "Wait, let me recheck my solution."
4. **Corrected Solution:** Introduced with green text: "Let me try again."
5. **Final Verification:** Another green-text section confirming the corrected answer.
### Detailed Analysis / Content Details
**1. Problem Statement:**
* **Text:** "Problem: There are an infinite number of vectors **v** such that `(1, 2, -5) × v = (90, 30, 30)`. Find the vector **v** that has the smallest magnitude."
**2. Initial (Incorrect) Solution Attempt:**
* **Method:** Sets `v = (x, y, z)`. Computes the cross product `(1, 2, -5) × (x, y, z)` as `(2z + 5y, -5x + z, 2x - y)`.
* **System of Equations:**
```
2z + 5y = 90
-5x + z = 30
2x - y = 30
```
* **Solving Process:** Solves the third equation for `y = 2x - 30`. Substitutes into the first to get `z = 120 - 5x`. Substitutes into the second to find `x = 9`. Then finds `y = -12` and `z = 75`.
* **Proposed Vector:** `v = (9, -12, 75)`.
* **Minimization Logic:** States the general solution is `v = (9, -12, 75) + t(1, 2, -5)` for scalar `t`. Claims magnitude is minimized when `t=0`, yielding the vector `(9, -12, 75)`.
**3. First Verification (Identifies Error):**
* **Text (in green):** "Wait, let me recheck my solution."
* **Method:** Uses the property that the cross product `a × b` is orthogonal to both `a` and `b`. Here, `a = (1, 2, -5)` and the result `c = (90, 30, 30)`. Therefore, the correct `v` must be orthogonal to `c`.
* **Check:** Computes dot products:
* `a · c = (1)(90) + (2)(30) + (-5)(30) = 90 + 60 - 150 = 0` (Orthogonal, as expected).
* `v · c = (9)(90) + (-12)(30) + (75)(30) = 810 - 360 + 2250 = 2700 ≠ 0`.
* **Conclusion (in red):** "Since `v · c ≠ 0`, the given answer **v** is not orthogonal to **c**, which means it cannot be the correct vector **v** that satisfies the cross product equation. Therefore, the answer is incorrect."
**4. Corrected Solution:**
* **Text (in green):** "Let me try again."
* **Method:** Recomputes the cross product using the determinant formula with unit vectors **i, j, k**:
```
| i j k |
| 1 2 -5 | = i(2z + 5y) - j(z + 5x) + k(y - 2x)
| x y z |
```
This yields the vector: `(2z + 5y, -z - 5x, y - 2x)`.
* **Correct System of Equations:**
```
2z + 5y = 90
-z - 5x = 30
y - 2x = 30
```
* **Solving Process:**
* From second equation: `z = -5x - 30`.
* From third equation: `y = 2x + 30`.
* Substitutes into first equation: `2(-5x - 30) + 5(2x + 30) = 90` → `-10x - 60 + 10x + 150 = 90` → `90 = 90`.
* **Key Insight:** The system is consistent and dependent, yielding infinitely many solutions. The general solution is expressed as:
`v = (x, 2x + 30, -5x - 30)`.
* **Minimization:** To find the smallest magnitude, minimizes `||v||² = x² + (2x+30)² + (-5x-30)²`.
* Simplifies to: `30x² + 420x + 1800`.
* Completes the square: `30(x + 7)² + 330`.
* The minimum occurs at `x = -7`.
* **Final Vector:** Substitutes `x = -7` into the general solution:
`v = (-7, 2(-7)+30, -5(-7)-30) = (-7, 16, 5)`.
**5. Final Verification:**
* **Text (in green):** "Wait, let me recheck my solution."
* **Method:** Repeats the orthogonality check with the new vector `v = (-7, 16, 5)`.
* **Check:**
* `a · c = 0` (as before).
* `v · c = (-7)(90) + (16)(30) + (5)(30) = -630 + 480 + 150 = 0`.
* **Conclusion:** "Since both dot products are zero, the given answer **v** is orthogonal to **c**, which means it could be a correct vector **v** that satisfies the cross product equation. Therefore, the answer is correct."
### Key Observations
1. **Error Identification:** The initial solution made a sign error in the second component of the cross product (`-5x + z` instead of `-z - 5x`). This was caught by applying the orthogonality property of the cross product.
2. **System Dependency:** The correct system of equations is linearly dependent, leading to a line of solutions parameterized by `x`.
3. **Minimization Technique:** The problem of finding the smallest magnitude vector is solved by minimizing the squared magnitude function, which reduces to finding the vertex of a parabola.
4. **Visual Cues:** The use of **green text** for self-correction and **red text** for the error statement provides clear visual segmentation of the problem-solving narrative.
### Interpretation
This document is a pedagogical walkthrough of a vector algebra problem. It demonstrates not just the solution, but the **process of mathematical reasoning and verification**. The core takeaway is the application of two key concepts:
1. **Geometric Property of Cross Product:** The result `a × b` is orthogonal to both `a` and `b`. This property serves as a powerful check for the correctness of a solution.
2. **Optimization in Linear Systems:** When a system has infinite solutions (a line), the solution with the minimum norm (smallest magnitude) is the one orthogonal to the direction vector of the line. The algebraic method of completing the square on the magnitude function is shown to find this point.
The narrative structure—attempt, error, correction, verification—mirrors authentic problem-solving and emphasizes the importance of checking work using fundamental properties. The final answer, `v = (-7, 16, 5)`, is the unique point on the solution line closest to the origin.
