Key Concepts and Formulas

1. Vector Cross Product Property: If $\vec{X} \times \vec{Y} = \vec{Z} \times \vec{Y}$ and $\vec{Y} \neq \vec{0}$, then $(\vec{X} - \vec{Z}) \times \vec{Y} = \vec{0}$. This implies that $\vec{X} - \vec{Z}$ is parallel to $\vec{Y}$, meaning $\vec{X} - \vec{Z} = \lambda \vec{Y}$ for some scalar $\lambda$.
2. Vector Dot Product Properties:
   • Distributive Property: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$.
   • Geometric Property: $(\vec{B} + \vec{C}) \cdot (\vec{B} - \vec{C}) = |\vec{B}|^2 - |\vec{C}|^2$.
3. Magnitude of a Vector: For a vector $\vec{V} = x\hat{i} + y\hat{j} + z\hat{k}$, its squared magnitude is $|\vec{V}|^2 = x^2 + y^2 + z^2 = \vec{V} \cdot \vec{V}$.
4. Scalar Multiplication of Magnitude: For a scalar $k$ and vector $\vec{V}$, $|k\vec{V}|^2 = k^2 |\vec{V}|^2$.

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Problem Statement

Given vectors $\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}$, $\vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$, and $\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$. Find the value of $\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}$, where $\vec{r}$ satisfies:

1. $\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}$
2. $\vec{r} \cdot(\vec{b}-\vec{c})=0$

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Step-by-Step Solution

Step 1: Determine the form of vector $\vec{r}$ using the first condition. The first condition is $\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}$. Rearranging this equation, we get: $\vec{r} \times \vec{a} - (\vec{b}+\vec{c}) \times \vec{a} = \vec{0}$ Using the distributive property of the cross product, we can factor out $\vec{a}$: $(\vec{r} - (\vec{b}+\vec{c})) \times \vec{a} = \vec{0}$ Since $\vec{a}$ is a non-zero vector, this implies that the vector $(\vec{r} - (\vec{b}+\vec{c}))$ must be parallel to $\vec{a}$. Therefore, we can express this relationship as: $\vec{r} - (\vec{b}+\vec{c}) = \lambda \vec{a}$ for some scalar $\lambda$. This gives us an expression for $\vec{r}$: $\vec{r} = (\vec{b}+\vec{c}) + \lambda \vec{a} \quad \cdots (1)$

Step 2: Use the second condition to find the scalar $\lambda$. The second condition is $\vec{r} \cdot(\vec{b}-\vec{c})=0$. Substitute the expression for $\vec{r}$ from equation (1) into this condition: $((\vec{b}+\vec{c}) + \lambda \vec{a}) \cdot (\vec{b}-\vec{c}) = 0$ Apply the distributive property of the dot product: $(\vec{b}+\vec{c}) \cdot (\vec{b}-\vec{c}) + \lambda (\vec{a} \cdot (\vec{b}-\vec{c})) = 0$ Using the identity $(\vec{X}+\vec{Y}) \cdot (\vec{X}-\vec{Y}) = |\vec{X}|^2 - |\vec{Y}|^2$, the first term becomes: $|\vec{b}|^2 - |\vec{c}|^2 + \lambda (\vec{a} \cdot (\vec{b}-\vec{c})) = 0 \quad \cdots (2)$ Now, we compute the necessary vector components and their magnitudes/dot products: $\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}$ $\vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$ $\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$

Calculate $|\vec{b}|^2$: $|\vec{b}|^2 = (3)^2 + (7)^2 + (-13)^2 = 9 + 49 + 169 = 227$ Calculate $|\vec{c}|^2$: $|\vec{c}|^2 = (17)^2 + (-2)^2 + (1)^2 = 289 + 4 + 1 = 294$ Calculate the difference of squares: $|\vec{b}|^2 - |\vec{c}|^2 = 227 - 294 = -67$ Calculate the vector $\vec{b}-\vec{c}$: $\vec{b}-\vec{c} = (3-17)\hat{i} + (7-(-2))\hat{j} + (-13-1)\hat{k} = -14\hat{i} + 9\hat{j} - 14\hat{k}$ Calculate the dot product $\vec{a} \cdot (\vec{b}-\vec{c})$: $\vec{a} \cdot (\vec{b}-\vec{c}) = (9)(-14) + (-13)(9) + (25)(-14)$ $= -126 - 117 - 350 = -593$ Substitute these values back into equation (2): $-67 + \lambda (-593) = 0$ $-593 \lambda = 67$ $\lambda = -\frac{67}{593}$

Step 3: Evaluate the target expression $\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}$. First, simplify the term $593 \vec{r}+67 \vec{a}$ by substituting the expression for $\vec{r}$ from equation (1) and the value of $\lambda$: $593 \vec{r}+67 \vec{a} = 593 ((\vec{b}+\vec{c}) + \lambda \vec{a}) + 67 \vec{a}$ $= 593 (\vec{b}+\vec{c}) + 593 \lambda \vec{a} + 67 \vec{a}$ $= 593 (\vec{b}+\vec{c}) + (593 \lambda + 67) \vec{a}$ Substitute $\lambda = -\frac{67}{593}$: $593 \lambda + 67 = 593 \left(-\frac{67}{593}\right) + 67 = -67 + 67 = 0$ So, the term simplifies to: $593 \vec{r}+67 \vec{a} = 593 (\vec{b}+\vec{c}) + (0) \vec{a} = 593 (\vec{b}+\vec{c})$ Now substitute this into the expression to be evaluated: $\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2} = \frac{|593 (\vec{b}+\vec{c})|^2}{(593)^2}$ Using the property $|k\vec{V}|^2 = k^2 |\vec{V}|^2$: $\frac{(593)^2 |(\vec{b}+\vec{c})|^2}{(593)^2} = |(\vec{b}+\vec{c})|^2$ Finally, calculate $\vec{b}+\vec{c}$ and its squared magnitude: $\vec{b}+\vec{c} = (3+17)\hat{i} + (7-2)\hat{j} + (-13+1)\hat{k}$ $\vec{b}+\vec{c} = 20\hat{i} + 5\hat{j} - 12\hat{k}$ Calculate the magnitude squared: $|(\vec{b}+\vec{c})|^2 = (20)^2 + (5)^2 + (-12)^2 = 400 + 25 + 144 = 569$

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Common Mistakes & Tips

• Cross Product Interpretation: Be precise with the implication of $(\vec{X} - \vec{Z}) \times \vec{Y} = \vec{0}$. It means $\vec{X} - \vec{Z}$ is parallel to $\vec{Y}$, not necessarily that $\vec{X} - \vec{Z} = \vec{0}$ or $\vec{X} = \vec{Z}$.
• Algebraic Simplification: The problem is designed for the term $(593\lambda + 67)$ to become zero. Always look for such cancellations, which often indicate a correct path or a simplified solution.
• Component-wise Operations: Ensure accuracy when performing vector addition, subtraction, and dot products. A single arithmetic error in components can lead to an incorrect final answer.

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Summary

The problem is solved by first using the cross product condition to establish the general form of $\vec{r}$ as $\vec{r} = (\vec{b}+\vec{c}) + \lambda \vec{a}$. The dot product condition is then used to solve for the scalar $\lambda$, which involves calculating the magnitudes of $\vec{b}$ and $\vec{c}$, and the dot product of $\vec{a}$ with $(\vec{b}-\vec{c})$. Finally, substituting the derived $\vec{r}$ and the value of $\lambda$ into the target expression $\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}$ leads to a significant simplification, where the terms involving $\lambda$ cancel out, leaving the result as $|(\vec{b}+\vec{c})|^2$. The calculation of this magnitude squared yields the final answer.

The final answer is $\boxed{569}$.