Let `veca, vecb, and vecc` be three non- coplanar vectors and `vecd` be a non -zero , which is perpendicular to `(veca + vecb + vecc). Now vecd = (veca xx vecb) sin x + (vecb xx vecc) cos y + 2 (vecc xx veca) `. Then

Options

A. `(vecd. (veca + vecc))/([veca vecb vecc])=2`
B. `(vecd. (veca + vecc))/([veca vecb vecc])=-2`
C. minimum value of `x^(2) + y^(2) is pi^(2)//4`
D. minimum value of `x^(2) + y^(2) is 5 pi^(2)//4`

Explanation

To solve the given problem step by step, we will analyze the provided information and derive the necessary relationships.

### Step 1: Understand the given vectors and their properties
We have three non-coplanar vectors \( \vec{a}, \vec{b}, \vec{c} \) and a vector \( \vec{d} \) which is perpendicular to the sum of these vectors:
\[ \vec{d} \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]

### Step 2: Express \( \vec{d} \)
The vector \( \vec{d} \) is given as:
\[ \vec{d} = (\vec{a} \times \vec{b}) \sin x + (\vec{b} \times \vec{c}) \cos y + 2(\vec{c} \times \vec{a}) \]

### Step 3: Substitute \( \vec{d} \) into the perpendicularity condition
We need to substitute \( \vec{d} \) into the condition that it is perpendicular to \( \vec{a} + \vec{b} + \vec{c} \):
\[ \vec{d} \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]
This expands to:
\[ [(\vec{a} \times \vec{b}) \sin x + (\vec{b} \times \vec{c}) \cos y + 2(\vec{c} \times \vec{a})] \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]

### Step 4: Expand the dot product
We will expand the dot product:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{a} + \vec{b} + \vec{c}) \sin x + (\vec{b} \times \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) \cos y + 2(\vec{c} \times \vec{a}) \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]

### Step 5: Evaluate each term
1. **First term**:
\[ (\vec{a} \times \vec{b}) \cdot \vec{a} = 0 \quad \text{and} \quad (\vec{a} \times \vec{b}) \cdot \vec{b} = 0 \]
Thus,
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c}) \sin x \]

2. **Second term**:
\[ (\vec{b} \times \vec{c}) \cdot \vec{a} + 0 + (\vec{b} \times \vec{c}) \cdot \vec{c} = 0 \]
Thus,
\[ (\vec{b} \times \vec{c}) \cdot \vec{a} \cos y \]

3. **Third term**:
\[ 2(\vec{c} \times \vec{a}) \cdot \vec{a} = 0 \quad \text{and} \quad 2(\vec{c} \times \vec{a}) \cdot \vec{b} + 2(\vec{c} \times \vec{a}) \cdot \vec{c} = 0 \]
Thus,
\[ 2(\vec{c} \times \vec{a}) \cdot \vec{b} \]

### Step 6: Combine the results
Combining all the terms, we have:
\[ (\vec{a} \times \vec{b}) \cdot \vec{c} \sin x + (\vec{b} \times \vec{c}) \cdot \vec{a} \cos y + 2(\vec{c} \times \vec{a}) \cdot \vec{b} = 0 \]

### Step 7: Solve for \( \sin x \) and \( \cos y \)
From the equation, we can express \( \sin x \) and \( \cos y \) in terms of the dot products of the vectors.

### Step 8: Find the minimum value
To find the minimum value of \( x^2 + y^2 \), we need to find values of \( x \) and \( y \) that satisfy the conditions derived above.

1. From the derived equations, we can set \( \sin x = -1 \) and \( \cos y = -1 \).
2. This gives \( x = -\frac{\pi}{2} \) and \( y = \pi \).

### Step 9: Calculate \( x^2 + y^2 \)
\[ x^2 + y^2 = \left(-\frac{\pi}{2}\right)^2 + \pi^2 = \frac{\pi^2}{4} + \pi^2 = \frac{5\pi^2}{4} \]

### Conclusion
Thus, the minimum value of \( x^2 + y^2 \) is \( \frac{5\pi^2}{4} \).

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