If `veca .vecb =beta and veca xx vecb = vecc ," then " vecb` is

Options

A. `((betaveca-vecaxxvecc))/(|veca|^(2))`
B. `((betaveca+vecaxxvecc))/(|veca|^(2))`
C. `((betavecc + vecaxxvecc))/(|veca|^(2))`
D. `((betavecc + vecaxxvecc))/(|veca|^(2))`

Explanation

To solve the problem step by step, we will use the properties of dot and cross products of vectors.

**Given:**
1. \(\vec{a} \cdot \vec{b} = \beta\)
2. \(\vec{a} \times \vec{b} = \vec{c}\)

**We need to find \(\vec{b}\).**

### Step 1: Use the vector triple product identity
We can use the vector triple product identity, which states:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]
In our case, we can rewrite \(\vec{a} \times \vec{b}\) as \(\vec{c}\) and apply the identity:
\[ \vec{a} \times (\vec{a} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{a}) \vec{b} \]

### Step 2: Substitute known values
Substituting the known values into the equation:
\[ \vec{a} \times \vec{c} = (\vec{a} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{a}) \vec{b} \]
Since \(\vec{a} \cdot \vec{b} = \beta\), we substitute this into the equation:
\[ \vec{a} \times \vec{c} = \beta \vec{a} - |\vec{a}|^2 \vec{b} \]

### Step 3: Rearranging the equation
Rearranging the equation to isolate \(\vec{b}\):
\[ |\vec{a}|^2 \vec{b} = \beta \vec{a} - \vec{a} \times \vec{c} \]
Now, divide both sides by \(|\vec{a}|^2\):
\[ \vec{b} = \frac{\beta \vec{a} - \vec{a} \times \vec{c}}{|\vec{a}|^2} \]

### Final Result
Thus, the expression for \(\vec{b}\) is:
\[ \vec{b} = \frac{\beta \vec{a} - \vec{a} \times \vec{c}}{|\vec{a}|^2} \]