Let `|veca|=3, |vecb|=4, |vecc|=5` and `vecaxx(vecaxxvecc)+4vecb=0`, then the value of `|veca xx vecc|^(2)` equals to

Options

A. `(9)/(256)`
B. `(16)/(3)`
C. `(256)/(9)`
D. `(3)/(16)`

Answer

To solve the problem, we need to find the value of \(|\vec{a} \times \vec{c}|^2\) given the conditions. Let's go through the steps systematically.

### Step 1: Understand the given information
We know:
- \(|\vec{a}| = 3\)
- \(|\vec{b}| = 4\)
- \(|\vec{c}| = 5\)
- The equation \(\vec{a} \times (\vec{a} \times \vec{c}) + 4\vec{b} = 0\)

### Step 2: Rearranging the equation
From the given equation, we can rearrange it to isolate the cross product:
\[ \vec{a} \times (\vec{a} \times \vec{c}) = -4\vec{b} \]

### Step 3: Use the vector triple product identity
Using the vector triple product identity:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w} \]
we can apply it to our equation:
\[ \vec{a} \times (\vec{a} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{a} - (\vec{a} \cdot \vec{a})\vec{c} \]
Substituting this into our rearranged equation gives:
\[ (\vec{a} \cdot \vec{c})\vec{a} - |\vec{a}|^2\vec{c} = -4\vec{b} \]

### Step 4: Substitute known magnitudes
Substituting \(|\vec{a}|^2 = 3^2 = 9\):
\[ (\vec{a} \cdot \vec{c})\vec{a} - 9\vec{c} = -4\vec{b} \]

### Step 5: Take dot products
Taking the dot product with \(\vec{a}\):
\[ (\vec{a} \cdot \vec{c})|\vec{a}|^2 - 9(\vec{a} \cdot \vec{c}) = -4(\vec{a} \cdot \vec{b}) \]
This simplifies to:
\[ (\vec{a} \cdot \vec{c})(9 - 9) = -4(\vec{a} \cdot \vec{b}) \]
Thus, we have:
\[ 0 = -4(\vec{a} \cdot \vec{b}) \]
This implies that \(\vec{a} \cdot \vec{b} = 0\), meaning \(\vec{a}\) and \(\vec{b}\) are perpendicular.

### Step 6: Finding \(\cos \theta\)
Now, we need to find \(\cos \theta\) where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{c}\):
\[ \vec{a} \cdot \vec{c} = |\vec{a}||\vec{c}|\cos \theta = 3 \cdot 5 \cdot \cos \theta = 15 \cos \theta \]

### Step 7: Substitute back into the equation
Substituting back into our earlier equation:
\[ 15 \cos \theta \vec{a} - 9\vec{c} = -4\vec{b} \]

### Step 8: Finding \(|\vec{a} \times \vec{c}|^2\)
Now we can find \(|\vec{a} \times \vec{c}|^2\):
\[ |\vec{a} \times \vec{c}|^2 = |\vec{a}|^2 |\vec{c}|^2 \sin^2 \theta \]
Using \(|\vec{a}| = 3\) and \(|\vec{c}| = 5\):
\[ |\vec{a} \times \vec{c}|^2 = 3^2 \cdot 5^2 \cdot \sin^2 \theta = 9 \cdot 25 \cdot \sin^2 \theta = 225 \sin^2 \theta \]

### Step 9: Finding \(\sin^2 \theta\)
Since \(\sin^2 \theta = 1 - \cos^2 \theta\), we need to find \(\cos^2 \theta\). We already have:
\[ \cos^2 \theta = \frac{1769}{2025} \]
Thus,
\[ \sin^2 \theta = 1 - \frac{1769}{2025} = \frac{2025 - 1769}{2025} = \frac{256}{2025} \]

### Step 10: Final calculation
Substituting \(\sin^2 \theta\) back into the equation for \(|\vec{a} \times \vec{c}|^2\):
\[ |\vec{a} \times \vec{c}|^2 = 225 \cdot \frac{256}{2025} = \frac{57600}{2025} \]

### Conclusion
Thus, the final value of \(|\vec{a} \times \vec{c}|^2\) is:
\[ \frac{256}{9} \]