If `veca and vecb ` are any two vectors of magnitude 2 and 3 respectively such that `|2(vecaxxvecb)|+|3(veca.vecb)|=k` then the maximum value of k is (a) `sqrt13` (b) `2sqrt13` (c) `6sqrt13` (d) `10sqrt13`

Options

A. `sqrt13`
B. `2sqrt13`
C. `6sqrt13`
D. `10sqrt13`

Explanation

To find the maximum value of \( k \) given the equation \( |2(\vec{a} \times \vec{b})| + |3(\vec{a} \cdot \vec{b})| = k \), we can follow these steps:

### Step 1: Understand the Magnitudes of the Vectors
We are given that the magnitudes of the vectors are:
- \( |\vec{a}| = 2 \)
- \( |\vec{b}| = 3 \)

### Step 2: Express the Cross Product and Dot Product
The magnitude of the cross product \( \vec{a} \times \vec{b} \) is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
where \( \theta \) is the angle between the vectors \( \vec{a} \) and \( \vec{b} \).

The dot product \( \vec{a} \cdot \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

### Step 3: Substitute the Magnitudes
Substituting the magnitudes into the equations gives:
\[ |\vec{a} \times \vec{b}| = 2 \cdot 3 \sin \theta = 6 \sin \theta \]
\[ \vec{a} \cdot \vec{b} = 2 \cdot 3 \cos \theta = 6 \cos \theta \]

### Step 4: Substitute into the Expression for \( k \)
Now we can substitute these into the expression for \( k \):
\[ k = |2(\vec{a} \times \vec{b})| + |3(\vec{a} \cdot \vec{b})| \]
This becomes:
\[ k = |2(6 \sin \theta)| + |3(6 \cos \theta)| \]
\[ k = 12 |\sin \theta| + 18 |\cos \theta| \]

### Step 5: Find the Maximum Value of \( k \)
To maximize \( k = 12 |\sin \theta| + 18 |\cos \theta| \), we can use the Cauchy-Schwarz inequality or the formula for the maximum value of \( A \sin \theta + B \cos \theta \):
\[ \text{Maximum value} = \sqrt{A^2 + B^2} \]
where \( A = 12 \) and \( B = 18 \).

Calculating this gives:
\[ \sqrt{12^2 + 18^2} = \sqrt{144 + 324} = \sqrt{468} = \sqrt{36 \cdot 13} = 6\sqrt{13} \]

### Conclusion
Thus, the maximum value of \( k \) is:
\[ k_{\text{max}} = 6\sqrt{13} \]

### Final Answer
The maximum value of \( k \) is \( 6\sqrt{13} \), which corresponds to option (c).

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