If `veca,vecb` are vectors perpendicular to each other and `|veca|=2, |vecb|=3, vecc xx veca=vecb`, then the least value of `2|vecc-veca|` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have two vectors, \(\vec{a}\) and \(\vec{b}\), which are perpendicular to each other. The magnitudes of these vectors are given as:
\[ |\vec{a}| = 2 \quad \text{and} \quad |\vec{b}| = 3 \]
Additionally, it is given that:
\[ \vec{c} \times \vec{a} = \vec{b} \]

### Step 2: Use the properties of cross products
Since \(\vec{c} \times \vec{a} = \vec{b}\), we can apply the formula for the magnitude of the cross product:
\[ |\vec{c} \times \vec{a}| = |\vec{c}| |\vec{a}| \sin \theta \]
where \(\theta\) is the angle between \(\vec{c}\) and \(\vec{a}\). Given that \(|\vec{b}| = 3\), we have:
\[ |\vec{c}| |\vec{a}| \sin \theta = 3 \]
Substituting \(|\vec{a}| = 2\):
\[ |\vec{c}| \cdot 2 \sin \theta = 3 \]
Thus, we can express \(|\vec{c}|\) in terms of \(\sin \theta\):
\[ |\vec{c}| = \frac{3}{2 \sin \theta} \]

### Step 3: Set up the expression to minimize
We need to find the least value of \(2 |\vec{c} - \vec{a}|\). Let \(K = |\vec{c} - \vec{a}|\). We can express \(K^2\) as:
\[ K^2 = |\vec{c}|^2 + |\vec{a}|^2 - 2 |\vec{c}| |\vec{a}| \cos \phi \]
where \(\phi\) is the angle between \(\vec{c}\) and \(\vec{a}\). Substituting the magnitudes:
\[ K^2 = \left(\frac{3}{2 \sin \theta}\right)^2 + 2^2 - 2 \left(\frac{3}{2 \sin \theta}\right) \cdot 2 \cos \phi \]
Simplifying:
\[ K^2 = \frac{9}{4 \sin^2 \theta} + 4 - \frac{6 \cos \phi}{\sin \theta} \]

### Step 4: Minimize the expression
To minimize \(K^2\), we can analyze the terms involving \(\sin \theta\) and \(\cos \phi\). We know that \(\sin^2 \theta + \cos^2 \phi = 1\). We can use this relationship to find the minimum value of \(K^2\).

### Step 5: Find the least value of \(2K\)
The minimum value of \(K\) occurs when the expression \(3 \cos \phi - 2 \sin \theta\) is minimized. Setting this expression equal to zero gives us the least value of \(K\). After calculations, we find:
\[ K = \frac{3}{2} \]
Thus, the least value of \(2K\) is:
\[ 2K = 2 \times \frac{3}{2} = 3 \]

### Final Answer
The least value of \(2 |\vec{c} - \vec{a}|\) is:
\[ \boxed{3} \]