Let `a =3i+2k and b=2j+k`. If c is a unit vector, then the maximum value of `[veca,vecb,vecc]` is :

Options

A. `sqrt(59)`
B. `sqrt(61)`
C. `sqrt(108)`
D. none

Answer

To solve the problem, we need to find the maximum value of the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\), where \(\vec{a} = 3\hat{i} + 2\hat{k}\) and \(\vec{b} = 2\hat{j} + \hat{k}\), and \(\vec{c}\) is a unit vector.

### Step 1: Calculate the Cross Product \(\vec{a} \times \vec{b}\)

The scalar triple product can be expressed as:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \times \vec{b} \cdot \vec{c} \]
First, we need to find \(\vec{a} \times \vec{b}\).

Using the determinant method:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 0 & 2 \\ 0 & 2 & 1 \end{vmatrix} \]

Calculating this determinant:
\[ = \hat{i} \begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & 2 \\ 0 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 0 \\ 0 & 2 \end{vmatrix} \]

Calculating the minors:
\[ = \hat{i} (0 \cdot 1 - 2 \cdot 2) - \hat{j} (3 \cdot 1 - 0 \cdot 2) + \hat{k} (3 \cdot 2 - 0 \cdot 0) \]
\[ = \hat{i} (-4) - \hat{j} (3) + \hat{k} (6) \]
\[ = -4\hat{i} - 3\hat{j} + 6\hat{k} \]

Thus, we have:
\[ \vec{a} \times \vec{b} = -4\hat{i} - 3\hat{j} + 6\hat{k} \]

### Step 2: Calculate the Magnitude of \(\vec{a} \times \vec{b}\)

Next, we calculate the magnitude of \(\vec{a} \times \vec{b}\):
\[ |\vec{a} \times \vec{b}| = \sqrt{(-4)^2 + (-3)^2 + (6)^2} \]
\[ = \sqrt{16 + 9 + 36} = \sqrt{61} \]

### Step 3: Find the Maximum Value of the Scalar Triple Product

The maximum value of the scalar triple product is given by:
\[ \text{Max}([ \vec{a}, \vec{b}, \vec{c} ]) = |\vec{a} \times \vec{b}| \cdot |\vec{c}| \cdot \cos(\theta) \]
Since \(\vec{c}\) is a unit vector, \(|\vec{c}| = 1\) and the maximum value of \(\cos(\theta)\) is 1 when \(\vec{c}\) is in the same direction as \(\vec{a} \times \vec{b}\).

Thus, we have:
\[ \text{Max}([ \vec{a}, \vec{b}, \vec{c} ]) = |\vec{a} \times \vec{b}| \cdot 1 = \sqrt{61} \]

### Final Answer
The maximum value of \([ \vec{a}, \vec{b}, \vec{c} ]\) is \(\sqrt{61}\).

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