Consider two vectors `veca=3hati-2hatj+4hatk and vecb=hatj+2hatk.` If `vecc` is a unit vector and k be the maximum value of `a.(vecbxxvecc)`, then the value of `k^(2)-50` is equal to

Answer

To solve the problem step by step, we will follow the mathematical operations and concepts involved in the scalar triple product.

### Step 1: Identify the vectors
We are given two vectors:
- \( \vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k} \)
- \( \vec{b} = \hat{j} + 2\hat{k} \)

We also know that \( \vec{c} \) is a unit vector.

### Step 2: Calculate the cross product \( \vec{b} \times \vec{c} \)
The scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) can be maximized. The maximum value occurs when \( \vec{c} \) is perpendicular to the plane formed by \( \vec{a} \) and \( \vec{b} \).

### Step 3: Calculate the magnitudes of vectors \( \vec{a} \) and \( \vec{b} \)
First, we find the magnitudes of \( \vec{a} \) and \( \vec{b} \):
- The magnitude of \( \vec{a} \):
\[ |\vec{a}| = \sqrt{3^2 + (-2)^2 + 4^2} = \sqrt{9 + 4 + 16} = \sqrt{29} \]

- The magnitude of \( \vec{b} \):
\[ |\vec{b}| = \sqrt{0^2 + 1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 4: Use the formula for the maximum value of the scalar triple product
The maximum value of the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) can be expressed as:
\[ k = |\vec{a}| \cdot |\vec{b}| \cdot |\vec{c}| \cdot \sin(\theta) \]
where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \). The maximum value occurs when \( \sin(\theta) = 1 \), which means \( \vec{c} \) is perpendicular to both \( \vec{a} \) and \( \vec{b} \).

Thus, we have:
\[ k = |\vec{a}| \cdot |\vec{b}| \cdot 1 = |\vec{a}| \cdot |\vec{b}| \]

### Step 5: Substitute the magnitudes
Substituting the magnitudes we found:
\[ k = \sqrt{29} \cdot \sqrt{5} = \sqrt{145} \]

### Step 6: Calculate \( k^2 - 50 \)
Now we need to calculate \( k^2 - 50 \):
\[ k^2 = (\sqrt{145})^2 = 145 \]
Thus,
\[ k^2 - 50 = 145 - 50 = 95 \]

### Final Answer
The value of \( k^2 - 50 \) is \( \boxed{95} \).